Math Easy Solutions

Math Easy Solutions

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In this video I go over an example on finding the volume by using the method of cylindrical shells. The example involves solving the volume of the solid formed by rotating the region bounded by the functions y = 2x^2 - x^3 and y = 0.

Download the notes in my video: http://1drv.ms/1IKEi9H

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-volumes-by-cylindrical-shells-example-1

Related Videos:

Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over another example on solving the volume of a shape using the method of cylindrical shells. The example in this video is finding the volume of the solid formed by rotating about the y-axis the region between y = x and x = x^2. In my earlier videos I solved this example using other methods but this time I show how it is easier to use the method of cylindrical shells to find the volume.

Download the notes in my video: http://1drv.ms/1xsRxce

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-volumes-using-cylindrical-shells-example-2

Related Videos:

Integrals and Volumes by Cylindrical Shells: Example 1: http://youtu.be/3n6xQqNzN74
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over another example on solving volumes of solids using the method cylindrical shells. This example involves solving the volume of the solid obtained by rotating about the x-axis the region under the curve y = sqrt(x) from x = 0 to 1.

Download the notes in my video: http://1drv.ms/1BgVGNW

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-volumes-by-cylindrical-shells-example-3

Related Videos:

Integrals and Volumes by Cylindrical Shells: Example 2: http://youtu.be/hR-GcP-XpD4
Integrals and Volumes by Cylindrical Shells: Example 1: http://youtu.be/3n6xQqNzN74
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In #MESExperiments 21 I go over a discovery I made early last year which flipped my already flipped understanding of gyroscopes, and that is adding extra weight to a gyroscope can make it rise faster. In some cases, this experiment being one of them, the rate of rising can be significantly faster. This video compares the rising rate of a 100.63 gram toy gyroscope with and without an extra 29.67 g putty weight. Initially both gyros start off at about 48° and somehow the weight added gyro rises about 2.71 times faster; 14 seconds vs. 38 seconds. Gyros rising on their own are amazing enough, but the ability for some gyroscopes (not all as future experiments will show) to rise much faster with relatively heavy weight added quite literally flies in the face of the conventional view of gravity.

Shouldn't gravity make things go down not up?! ;)

The original unedited experiments are shown in the links below:

- No weight added test: https://youtu.be/GUFSjbiLFw4
- Weight added test: https://youtu.be/agBSVjmE-nM

The full experiment results are tabulated, graphed, and presented on Steemit: https://steemit.com/mesexperiments/@mes/mesexperiments-21-added-weight-can-make-a-gyroscope-rise-faster-magic

Although the weight added gyro rose much faster, it also had a much shorter “sleep” period and also fell down much sooner too. Interestingly, the unweighted gyro took about 2.7 times the time to rise as well as 2.7 times the overall time before fall than that of the weighted gyro. This raises some possible relationship between rising rate and overall balancing time.

Note that the weighted gyro has significantly more “rotational inertia” from the non-spinning extra mass which effectively throws the gyro off the support resulting in a shorter sleep and overall shorter balancing period.

The full sequence of events and weight measurements are also presented on Steemit.

Stay Tuned for #MESExperiments 22…

Related Videos:

🔬#MESExperiments 11: Increasing Gyroscope Spin Speed Doesn't Necessarily Increase Rising Rate: https://steemit.com/mesexperiments/@mes/mesexperiments-11-increasing-gyroscope-spin-speed-doesn-t-necessarily-increase-rising-rate

🔥#MESScience 1: How Does a Powerball Gyroscope Work? + Gyros Are Inverted Pendulums: https://steemit.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums

☝#AntiGravity Part 6: Objects in Rotation Defy ‘Mainstream’ Physics + MES Duality Concept: https://steemit.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept

#MESExperiments Video Series: https://steemit.com/mesexperiments/@mes/list
DRAFT #MESExperiments Video Series: https://mes.fm/experiments-draft
#MESScience Video Series: https://steemit.com/science/@mes/tutorials
#AntiGravity Video Series: https://steemit.com/antigravity/@mes/series
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist .

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In this video I go over another example on solving volumes using cylindrical shells. In this example I solve for the volume of the solid formed by rotating the region bounded by y = x - x^2 and y = 0 about the line x = 2.

Download the notes in my video: http://1drv.ms/1F2iJly

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-volumes-by-cylindrical-shells-example-4

Related Videos:

Integrals and Volumes by Cylindrical Shells: Example 3: http://youtu.be/jRZUoJsPRC4

Integrals and Volumes by Cylindrical Shells: Example 2: http://youtu.be/hR-GcP-XpD4
Integrals and Volumes by Cylindrical Shells: Example 1: http://youtu.be/3n6xQqNzN74
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over the concept of work in terms of the definition of work as the force multiplied by displacement (W = F·d) as well as defining it using integrals as the area under the curve of a force, f(x), function where x is the position. I also go over the units of work which are Joules (newton-meters) in SI Units and in Foot-Pounds in US Customary units.

Download the notes in my video: http://1drv.ms/1xKz3SJ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-work-introduction

Related Videos:

Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over 2 examples on determining the amount of work or energy required in performing an action. In this video first example the force is constant, which is the force of gravity acting on a book. In the second example the force is variable and thus the amount of Work is calculated by using integrals as the summation of the area of under the force function.

Download the notes in my video: http://1drv.ms/1AGgZsR

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-work-example-1

Related Videos:

Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over another example on work and this time deal with Hooke's Law. Hooke's Law states that the force required to maintain a spring stretched x units beyond its natural length is proportional to x and can be written as f(x) = kx where k is a positive constant and is known as the spring constant. The example that I cover is an example involving stretching a spring and determining the amount of work that was required.

Download the notes in my video: http://1drv.ms/1xWkRGp

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-work-example-2-hooke-s-law

Related Videos:

Integrals and Work Example 1: http://youtu.be/YO0M0w12URo
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over pivot tables and pivot charts in Microsoft Excel. Pivot Tables are an extremely powerful tool to analyze and modify lots of data very quickly and easily. Being able to use pivot tables is a very important skill to have to make sure to watch this video!

Download the Excel File in my video: http://1drv.ms/1BEa1X8

Related Videos:

Convert Time to number, seconds, hours and minutes in Microsoft Excel 2007: http://youtu.be/fwMKbXI98tI
Converting Numbers to Fractions in Microsoft Excel 2007: http://youtu.be/YgJBK0QjIec
Intro to Tables in Excel 2007: http://youtu.be/B_zXmunqrl4
Drop Down Lists with Microsoft Excel 2007: http://youtu.be/e7Sbcn-y9j4
Microsoft Excel Tips: -2^2 = +4 and NOT -4!!!: http://youtu.be/9QC3ZboI3wo
Skip Blanks when Graphing in Microsoft Excel: http://youtu.be/HcCJVUd7a5Q
Transparent Images in Microsoft Excel + Useful Tip: http://youtu.be/uKjtsQZ5Ow0
Automate Formulas using INDIRECT() Function in Excel: http://youtu.be/9WiBtrNq9yU
ISERROR and ISERR Functions in Microsoft Excel: http://youtu.be/MrZLtpsLktM
Trendline Slope and Intercept Formulas in Microsoft Excel: http://youtu.be/DTDY9RL35b8
Simplfying Formulas in Automated Tables in Excel: http://youtu.be/gFhT9OD_OyU
Remove Trendline from the Legend in Microsoft Excel: http://youtu.be/njp-zx_5Dq0
Converting Text to a Table using "Fixed Width Text to Columns" in Microsoft Excel: http://youtu.be/wOpcnsMHb6Y
Adding Secondary Vertical and Horizontal Axes in Microsoft Excel: http://youtu.be/fhPyaWzIQVk
Microsoft Excel: Graphing too many data points - Simple Solution: http://youtu.be/jKwFDR5X1ZY
Microsoft Excel: Data Bars within Cells: http://youtu.be/EjvAhSLmJgw
Substitute Function in Microsoft Excel: http://youtu.be/E9vWWWAqSvw .

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In #MESExperiments 20 I’ve uploaded one of the least known experiments by the late great Eric Laithwaite and it is among the clearest demonstrations of gyroscopic weight loss through forced precession. This experiment involves placing a gyro wheel with a shaft that is loosely connected onto a vertical rod which allows the gyro and shaft setup to freely move upwards if the gyro “chooses” to. And choosing to rise upwards is exactly what it does as Eric forces the precession with an external looped structure which he pushes against the end of the shaft.

As the gyro torques upwards and thus the center of mass increases so too does the entire gyro + shaft setup rise; and in fact rise completely airborne for a brief moment. Thus, had there been a weight scale underneath, it would measure a zero weight reading as it was airborne; in other words, the gyro “loses weight” during forced precession, or more accurately, as the center of mass increases and without expected downwards reaction (which otherwise would increase the weight scale reading akin to a person pushing off the ground when jumping upwards).

This experiment is taken from the BBC “Heretics” series referenced below. Interestingly it is not available on YouTube, and BBC doesn’t allow anyone to upload it there either; thus raising the question of why the tax payer funded BBC, to the tune of about $5+ billion per year, places a cover on science. In fact even the video itself is a coverup of true gyro physics as it intermixes Eric's (conventional) physics defying experiments with nonsense commentary from the narrator and fake "scientists" that serve to merely confuse the public.

- Banned YouTube Video: https://www.youtube.com/watch?v=dy6hLZ02naM

Luckily I found an upload on Dailymotion and have also added a local video download link in case it gets removed there too.

- Dailymotion “Heretics” Video: http://www.dailymotion.com/video/x2ya0y4
- Local Download Link: https://1drv.ms/v/s!As32ynv0LoaIh5sxvfDT9KkkVF1Hog

Interestingly, this simple and clear-cut demonstration of gyroscopic weight loss is something that controlled fake science outlets such as the Veritasium YouTube channel (along with the BBC and Wikipedia et al) would rather have covered up; and which I explain in further detail in my world changing #AntiGravity Part 6 video: https://steemit.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept.

Also, I would briefly like to mention that, as explained in #MESScience 1: https://steemit.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums, that precession is always forced, hence a gyroscope when merely rising on its own like in many of my earlier experiments should also be losing weight as the center of mass rises. But in those cases the rate of rising is often too slow for most weight scales to detect.

The full experiment results can be viewed on Steemit: https://steemit.com/mesexperiments/@mes/mesexperiments-20-forced-precession-of-a-gyroscope-generates-inertial-lift-i-e-weight-loss.

Stay Tuned for #MESExperiments 21…

Related Videos:

🔥#MESScience 1: How Does a Powerball Gyroscope Work? + Gyros Are Inverted Pendulums: https://steemit.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums

☝#AntiGravity Part 6: Objects in Rotation Defy ‘Mainstream’ Physics + MES Duality Concept: https://steemit.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept

#MESExperiments Video Series: https://steemit.com/mesexperiments/@mes/list
DRAFT #MESExperiments Video Series: https://mes.fm/experiments-draft
#MESScience Video Series: https://steemit.com/science/@mes/tutorials
#AntiGravity Video Series: https://steemit.com/antigravity/@mes/series
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist .

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In #MESExperiments 19 I demonstrate that gyroscopes are a special form of inverted pendulums in that they don’t need any electronic feedback loop controls to balance or rise against gravity all on their own. This is one of the most fascinating capabilities of gyroscopes but is almost as fascinating and perplexing as how the entire world has missed out on this feat of gravity defying proportions. Although gyroscopes and spinning tops have always been inverted pendulums in principle, it is this particular experiment that shows that they are also inverted pendulums in form and appearance as well.

I demonstrate this inverted pendulum property of gyroscopes by showcasing two experiments involving a custom low friction gimbal setup on a string: the first with a light counterweight in which the gyroscope "forces its own precession" and the second with a heavy counterweight that requires manually forcing the precession initially.

The first experiment involves placing a light 6.60 gram counterweight with a 4.60 gram and 55 mm long stem onto a 148.68 gram gyroscope that is placed in a custom gimbal setup hung via a string to minimize rotational friction. The gyro in this case is able to raise the light counterweight on its own.

The second experiment, @ the 13:35 mark of this video, is similar to the first but involves a much heavier 148.68 gram counterweight. The weight is too much for this current gyro to raise on its own initially so I have manually torqued the gyro by slightly (i.e. torque over-unity) turning the gimbal to "hurry the precession" which magically causes the counterweight to rise upwards.

In both cases, and once in their risen inverted position, the gyroscope is able to magically balance the counterweights against gravity!

The original unedited versions of these experiments are listed below:

- Light counterweight test: https://youtu.be/PaD9tBYT7s4
- Heavy counterweight test: https://youtu.be/ubujSK_XiYE

In order for the gyroscope to lift the heavy counterweight on its own, a larger "spin angular momentum" (i.e. combination of larger mass and faster spin speed) is required; or just to lower the amount of counterweight torque by moving the string support more in the middle, as in #MESExperiment 9: https://steemit.com/mesexperiments/@mes/mesexperiments-9-gyroscope-with-counterweight-hung-on-a-string-rises-downwards-magicx1000

The gyro and counterweight weight and length measurements are shown in this video: https://youtu.be/LiZqJdraNvI

The full experiments results, discussion, and measurements are presented on Steemit: https://steemit.com/mesexperiments/@mes/mesexperiments-19-a-gyroscope-is-an-inverted-pendulum-without-electronic-sensors

For an overview of inverted pendulums, as well as discussion into what "precession" may truly be, make sure to watch my game-changing #MESScience 1 video: https://steemit.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums

Stay Tuned for #MESExperiments 20…

Related Videos:

🔥#MESScience 1: How Does a Powerball Gyroscope Work? + Gyros Are Inverted Pendulums: https://steemit.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums

🔬#MESExperiments 9: Gyroscope With Counterweight Hung on a String Rises 'DOWNWARDS'! #MagicX1000: https://steemit.com/mesexperiments/@mes/mesexperiments-9-gyroscope-with-counterweight-hung-on-a-string-rises-downwards-magicx1000

#MESExperiments Video Series: https://steemit.com/mesexperiments/@mes/list
DRAFT #MESExperiments Video Series: https://mes.fm/experiments-draft
#MESScience Video Series: https://steemit.com/science/@mes/tutorials
#AntiGravity Video Series: https://steemit.com/antigravity/@mes/series
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist .

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In this video I go over further into infinite sequences and series; and in particular expand upon my earlier video on power series. This time I show how we can represent many functions as power series as well as go over a theorem that allows for differentiation and integration of power series term by term. This theorem, as well as this overall concept of representing functions as power series is one of the most important concepts in all of science because it allows for much easier computation of many different functions. In fact many computers and calculators use this technique behind the scenes since they are often much easier to deal with than trying to find exact integrals or derivatives which many times are not simple.

In this video, and at the second half of this video, I also go over an extensive 2 hour long proof of the theorem for term-by-term differentiation and integration of power series. Although this proof is extensive, after searching all over online for the proof, I believe my proof may in fact be the most clear and understandable. Let me know if it helps!

The topics covered in this video are listed below with their time stamps.

1. @ 2:24 - Representation of Functions as Power Series
- @ 3:17 - Equation 1
- @ 10:48 - Example 1
- @ 26:35 - Example 2
- @ 30:16 - Example 3
2. - @ 37:06 - Differentiation and Integration of Power Series
- @ 38:09 - Theorem 1: Term by Term Differentiation and Integration
- @ 50:50 - Example 4
- @ 54:18 - Example 5
- @ 58:36 - Example 6
- @ 1:11:02 - Example 7: Gregory's Series
- @ 1:25:59 - Leibniz Formula for π
- @ 1:30:30 - Example 8
3. Exercises
- @ 2:11:57 - Exercise 1
- @ 1:55:34 - Exercise 2
- @ 2:26:11 - Exercise 3: Proof of Theorem 1
- @ 2:32:29 - Lemma
- @ 2:34:58 - Lemma Derivative Proof
- @ 3:01:16 - Lemma Integral Proof
- @ 3:08:59 - Theorem 1 Using z = (x - a)
- @ 3:12:41 - Theorem 1a Derivative Proof
- @ 3:24:59 - Brief Overview of the Binomial Theorem
- @ 3:57:33 - Theorem 1b Integral Proof

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_17muwohTwCBpQ45Q?e=pldHwj

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-representations-of-functions-as-power-series

Related Videos:

Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz

Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://steemit.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more

Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://steemit.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more

Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums

Infinite Sequences and Series: The Comparison Tests: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests

Infinite Sequences and Series: Alternating Tests: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests

Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests

Infinite Sequences and Series: Strategy for Testing Series: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series

Infinite Sequences and Series: Power Series: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-power-series .

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In this video I go over another example on determining the amount of work done in moving an object using integrals and this time I look at pulling a rope from the top of a building. In this example, the force required to pull the rope up is not given but we can determine the force and hence the work by breaking the rope into many small pieces and then use integration to determine the equation and hence amount of work required to pull up the rope. The reason we need to consider the rope as many small pieces is because as the rope is pulled up the length of the rope changes and thus the amount of force required keeps decreasing as the length of rope being pulled up decreases.

Download the notes in my video: http://1drv.ms/1CFoEb3

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-work-example-3-pulling-a-rope

Related Videos:

Integrals and Work: Example 2 - Hooke's Law: http://youtu.be/h8lwvB9rOng
Integrals and Work Example 1: http://youtu.be/YO0M0w12URo
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over another example on calculating work using integrals and solve for the amount of work required in pumping water out of an inverted circular cone.

Download the notes in my video: http://1drv.ms/1yNleVZ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-work-example-4-water-tank

Related Videos:

Integrals and Work: Example 3 - Pulling a Rope: http://youtu.be/Koy5vZxzR9U
Integrals and Work: Example 2 - Hooke's Law: http://youtu.be/h8lwvB9rOng
Integrals and Work Example 1: http://youtu.be/YO0M0w12URo
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over another video on the average value of a function and derive the formula for average value using integrals. This is important when averaging out an infinite amount of numbers as opposed to simple finite amount of numbers which require adding them up and dividing by the total amount of numbers.

Download the notes in my video: http://1drv.ms/1CH4Wvb

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-average-value-of-a-function

Related Videos:

Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: Absolute Value: http://youtu.be/w7HlDFcDNUs
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over an example on solving for the average value of a function and solve the average value of the function y = 1 + x^2 from x = - 1 and x = 2.

Download the notes in my video: http://1drv.ms/1BSQsf3

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-average-value-example-1

Related Videos:

Integrals and Average Value of a Function: http://youtu.be/Xo2SSQOki1g
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: Absolute Value: http://youtu.be/w7HlDFcDNUs
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over the Mean Value Theorem for Integrals which states that for any continuous function from x = a to x = b there exists a number c in that interval such that f(c) is equal to the average of the function during that interval. The theorem is a direct consequence of the original Mean Value Theorem which is for derivatives. This theorem also shows that in that interval the average value can be considered the height of a rectangle with the same area as that under the curve.

Download the notes in my video: http://1drv.ms/1JCnGP3

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/mean-value-theorem-for-integrals

Related Videos:

Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38
Integrals and Average Value: Example 1: http://youtu.be/Ps1IPP4O_Lo
Integrals and Average Value of a Function: http://youtu.be/Xo2SSQOki1g
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: Absolute Value: http://youtu.be/w7HlDFcDNUs
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over the proof of the mean value theorem for integrals which I covered in my last video. The proof considers a function written as an integral and by applying the original mean value theorem for derivatives the result will yield the mean value theorem for integrals which is very similar to the one for derivatives.

Download the notes in my video: http://1drv.ms/1uOBfcP

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/mean-value-theorem-for-integrals-proof

Related Videos:

Integrals and Average Value: Mean Value Theorem: http://youtu.be/rjI7ropMwuI
Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38
Integrals and Average Value: Example 1: http://youtu.be/Ps1IPP4O_Lo
Integrals and Average Value of a Function: http://youtu.be/Xo2SSQOki1g
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: Absolute Value: http://youtu.be/w7HlDFcDNUs
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over an example on using the mean value theorem for integrals to first show that there are 2 x-values that correspond to the average value of the function and then I solve for the x-values.

Download the notes in my video: http://1drv.ms/1uwwEGs

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/mean-value-theorem-for-integrals-example-1

Related Videos:

Mean Value Theorem for Integrals: Proof: http://youtu.be/HIhDyXM87pI
Integrals and Average Value: Mean Value Theorem: http://youtu.be/rjI7ropMwuI
Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38
Integrals and Average Value: Example 1: http://youtu.be/Ps1IPP4O_Lo
Integrals and Average Value of a Function: http://youtu.be/Xo2SSQOki1g
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: Absolute Value: http://youtu.be/w7HlDFcDNUs
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In #MESScience 1, I introduce my latest video series: MES Science Tutorials and continue on my pursuit for understanding true gyroscope physics. Since conventional science doesn’t meet the very high Math Easy Solutions certified (by me) standard, it was inevitable that I create this video series.

In the first installment of #MESScience, I take a look at the powerball gyroscope (generically referred to as a gyroscopic exercise tool), then shift gears to the realization that gyroscopes and spinning tops are inverted pendulums without electronic sensors or feedback loops, and then finally finish off with what “precession” may truly be.

This is a pretty epic video and it brings my understanding of gyroscopes, which I believe underpins the very foundation of all of science, closer to what is actually going on. This new understanding allows for the possibility of extending conventional physics beyond the mainstream in hopes of quantifying and engineering new applications and discoveries.

The topics covered in this video are listed below with their time stamps.

- @ 1:09 - Note on MES Hiatus From Uploading Videos
- @ 1:55 - Topics to Cover

1. @ 8:38 - Links to Other Series
2. @ 9:04 - Links to #MESScience Video Series
3. @ 9:29 - Introduction to #MESScience
4. @ 10:54 - Note on the Still Unexplained Gyroscope
5. @ 11:54 - The Powerball Gyroscope
6. @ 22:10 - Mainstream Explanation of the Powerball
- @ 44:04 - Quick Summary of Mainstream Powerball Physics
7. @ 46:07 - Recap of my #AntiGravity Part 6
- @ 57:10 - MES REALIZATION: Conventional "Angular Momentum" and Its "Conservation" are at Best "Relative" to Applied Force
- @ 1:05:51 - MES REALIZATION: Forced Precession Torque Over-Unity Depends on Amount of Forced Supplied
- @ 1:12:23 - Quick Summary of #AntiGravity Part 6 Findings
8. @ 1:14:15 - Effects of Rolling, Spin Speed, and Added Weight on Gyros and Spinning Tops
- @ 1:51:21 - Eric Laithwaite Had Unwittingly Demonstrated the Basic Mechanics of a Powerball in 1974
- @ 1:58:06 - MES REALIZATION: Precession of Spinning Tops Should Not Be Confused With Rolling Trajectory
- @ 2:04:06 - Quick Summary of Effects of Rolling, Spin Speed, and Added Weight
9. @ 2:05:45 - MES Breakdown of and Experiments with a Gyro Powerball
- @ 2:25:58 - Quick Summaries of MES Powerball Breakdown and MES Powerball Experiments
10. @ 2:28:50 - MES BOMBSHELL DISCOVERY: Added Weight Can Prevent Rising on Low Friction Surfaces
- @ 2:32:44 - Does Preventing Centrifugal Precession Condense into a Spiraling Upwards #Vortex Precession?
11. @ 2:36: 17 - MES BOMBSHELL REALIZATION: Gyroscopes and Spinning Tops are Inverted Pendulums Without Electronic Sensors
- @ 2:43:50 - Mathieu's Equation and My Previous Lifetime?? #MESDuality
- @ 2:49:25 - MES Custom Gimballed Gyroscope Inverted Pendulum
12. @ 3:05:05 - Quick Summary of Recent Bombshell Gyro Realizations
13. @ 3:08:35 - I THINK I SOLVED PRECESSION!
- @ 3:33:20 - MES BOMBSHELL REALIZATION: Precession is Merely Gyro Orientation Change Due to Applied Torque
- The Conventional View of Precession is Wrong!
- MES BOMBSHELL REALIZATIONS:
- @ 3:39:22 - Precession is a Torque and Moves the Ice Block
- @ 3:41:58 - Precession is Always “Forced”
- @ 3:56:05 - Spin Angular Momentum 'Absorbs' and Re-Directs Torque
- The Most Accurate Representation of Gyroscopic Anti-Gravity Motions
- @ 4:03:03 - Quick Summary of MES True Precession

**Stay Tuned for #MESScience 2…**

PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh_xmfIOk-PgRi0gdyg?e=8KXRlZ

Steemit Notes: https://steemit.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums

Related Videos:

#AntiGravity: https://steemit.com/antigravity/@mes/series
#FreeEnergy: https://mes.fm/freeenergy-playlist
#MESExperiments: https://steemit.com/mesexperiments/@mes/list
DRAFT #MESExperiments: https://mes.fm/experiments-draft
.

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In this video I go over an example on average value and show that the average velocity when defined as the change in position over the change in time is equal to the average of the average velocities or the average velocity function.

Download the notes in my video: http://1drv.ms/1zWoVc0

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integrals-and-average-value-example-2-velocity

Related Videos:

Mean Value Theorem for Integrals: Example 1: http://youtu.be/1rkqf6qVBXQ

Mean Value Theorem for Integrals: Proof: http://youtu.be/HIhDyXM87pI
Integrals and Average Value: Mean Value Theorem: http://youtu.be/rjI7ropMwuI
Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38
Integrals and Average Value: Example 1: http://youtu.be/Ps1IPP4O_Lo
Integrals and Average Value of a Function: http://youtu.be/Xo2SSQOki1g
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: Absolute Value: http://youtu.be/w7HlDFcDNUs
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over another Applied Project problem and go about answering the problem of determining the best seat to sit at when at the movie theatre. The applied project problems are from the end of each chapter in my Calculus textbook and they represent problems in everyday life that can be solved using Calculus. In this particular problem, the best seat to sit at is to maximize the viewing angle when watching a movie to ensure that the largest part of the movie screen is being seen at the most direct path. This problem is very useful for whenever you are contemplating which seat you should take, just like Rebecca Black ;)

Download the notes in my video:

Notes: http://1drv.ms/1tGGL0f
Excel Sheet: http://1drv.ms/1zJ1tzj
TI Nspire CAS file: http://1drv.ms/1zJ1Aee
Link to Download TI Nspire CAS Free Trial Software: http://tinyurl.com/op7s9ts

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applied-project-where-to-sit-at-the-movies

Related Videos:

Applied Project: Building a Better Roller Coaster: http://youtu.be/QjtaMIlKYxQ
Applied Project: Where Should a Pilot Start Descent?: http://youtu.be/DI4WMqrYB4Q
Applied Project: How Rainbows are Formed and the Order of it's Colors : http://youtu.be/Q3Euh4K93_c
Applied Project: How Double Rainbows are formed - Calculus of Rainbows Part 2 : http://youtu.be/bKfyV_YKLR4
Applied Project: The Shape of a Can - The Science behind Can Manufacturing : http://youtu.be/pfpd8RJi9jw
Integrals and Average Value of a Function: http://youtu.be/Xo2SSQOki1g .

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In this video I go over the method of integration by parts which allows the ability to simplify a more complex function to make integration easier. The integration by parts method corresponds to the product rule for derivatives in much the same way as the substitution rule corresponds to the chain rule for derivatives. This method is very important and is used throughout integral calculus so it is important to watch this video to fully understand the proof of it!

Download the notes in my video: http://1drv.ms/1yZoQm9

Related Videos:

Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk
The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over a pretty useful example on solving integrals by the method of integration by parts. The example I cover is the integral of the function x·sin(x) and show how to apply the integration by parts method as well as the simplified version using the u, du, v, dv method. I go into great detail in this example to make sure to illustrate the method of integration by parts as best as I can so make sure to watch this video!

Download the notes in my video: http://1drv.ms/1D7lkHW

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-parts-example-1

Related Videos:

Integration by Parts: Proof: http://youtu.be/TZhEOct5u_M
Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk
The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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In this video I go over another example on solving integrals using the method of Integration by Parts. The example involves solving the integral of ln(x).

Download the notes in my video: http://1drv.ms/1BYOchl

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-parts-example-2

Related Videos:

Integration by Parts: Example 1: http://youtu.be/K8pReMFOJl4
Integration by Parts: Proof: http://youtu.be/TZhEOct5u_M
Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk
The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .

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Created 2 years, 4 months ago.

545 videos

CategoryEducation

I mainly teach math, but also do controversial videos that are censored on YouTube, as well as creating #FreeEnergy technology!