Approximate Integration: Left, Right, Midpoint, and Trapezoidal Rules
In this video I go over the topic of approximate integration and explain how for the situations which the antiderivative can't be found or for experimental data where there is no formula, making an approximation of a definite integral is the only way to go. I go over 4 types of Riemann Sums Approximation methods: Left Endpoint, Right Endpoint, Midpoint and Trapezoidal Approximations methods. I went over Riemann Sums in my earlier video on definite integrals and showed that Riemann Sums are simply approximations of integrals using rectangles but when summed up to infinite they form the exact integral. In this video though, I focus on the 4 already mentioned approximation types and preview the fact that the Midpoint Rule is the most accurate approximation even though the Trapezoidal Rule appears at first glance as the most accurate. I explain this in detail in later videos so stay tuned!
Here is a simple "tutorial" on how to make a campfire ;) Actually it was just a random experiment I tried with gasoline in a campfire during my camping trip to Shuswap Lake on Canada Day long weekend. DON'T TRY THIS AT HOME.
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Cracking an Egg with One Hand: http://youtu.be/7R7P6Sehl9s
Trying to Convince My Younger Brother Markonnese to do a Math Easy Solution - BEDMAS: http://youtu.be/VLrK1tjqajE .
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https://www.youtube.com/watch?v=OH9U4RZ7VSo
In this video I go over a pretty extensive tutorial on infinite series, its definition, and many examples to elaborate in great detail. An infinite series is the summation of a sequence of terms as the terms approach an infinite number of terms and follow from my earlier video on infinite sequences; thus much of the previous video applies to this video. Several important series are covered such as the Geometric series and the Harmonic Series covered. Also, several interesting derivation techniques are utilized such as partial fraction decomposition and the amazing telescoping sum concept. I go over many examples and 2 important exercises from my calculus book (Early Transcendentals by James Stewart) to make the concept of infinite series, and their divergence or convergence, much more clear.
The topics and sections covered in this video are listed below with their timestamps:
- @ 0:00 - Intro
1. @ 1:34 - Infinite Series and Notation
- @ 8:29 - Definition 1
2. @ 13:11 - Example 1: Geometric Series
- @ 27:43 - Example 2
- @ 32:49 - Example 3
- @ 39:49 - Example 4
- @ 48:28 - Example 5
3. @ 51:10 - Example 6: Telescoping Sum
4. @ 1:01:31 - Example 7: Harmonic Series
5. @ 1:10:30 - Theorem 1
- @ 1:16:02 - Test for Divergence
- @ 1:17:30 - Example 8
6. @ 1:20:35 - Theorem 2
- @ 1:28:15 - Example 9
7. End of Chapter Exercises
- @ 1:35:10 - Exercise 1
- @ 1:45:43 - Exercise 2
Download the notes in my video:
PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh8gxUmX71Fk9dZ5_wg
Excel File: https://1drv.ms/x/s!As32ynv0LoaIh8gwtUIvFFz4HKE0DA
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more
Related Videos:
Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .
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In this video I go over Question 5 of the Discovery Project: Rotating on a Slant video series. In this question, I derive the surface area formula of the shape generated by rotating the curve about a slanted line. The derivation is fairly easy because the majority of the work was in Question 1 when the variables on the slanted line were rotated to write in terms of x.
Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!104392&authkey=!AM7ZI568EbxWse0&ithint=file%2cpdf
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/discovery-project-rotating-on-a-slant-question-5
Related Videos:
Discovery Project: Rotating on a Slant: Question 4: https://youtu.be/w3eR3d-dJRI
Discovery Project: Rotating on a Slant: Question 3: https://youtu.be/zwodt1OsFjE
Discovery Project: Rotating on a Slant: Question 3: https://youtu.be/zwodt1OsFjE
Discovery Project: Rotating on a Slant: Question 1: https://youtu.be/dXlhpDOBDe0
Discovery Project: Patterns in Integrals: Question 1: 1/((x+a)(x+b)): http://youtu.be/uQUu0XFMw8I
Discovery Project: Patterns in Integrals: Question 2: sin(ax)cos(bx): http://youtu.be/O0p1DGMOcfM
Discovery Project: Patterns in Integrals: Question 3: x^n * ln(x): http://youtu.be/PA9-Me1ko10
Discovery Project: Patterns in Integrals: Question 4: x^n * e^6: http://youtu.be/XgE75Pa0n-c
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU .
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https://www.youtube.com/watch?v=6c_4v9JeFsI
In this video I go over further into proving the parametric equations for the Cycloid curve and this time show that these equations do in fact work for all angles that the circle rotates to. In my earlier videos I derived the proof for the cases where θ is between 0 and π/2, as well as between π/2 and π. In this video I first show that the equations still work for the values θ = 0, π/2, and pi. From there I show that it still works for all other values of θ because of the cyclical nature of the trigonometric functions in the cycloid equations. Thus as the values of θ change, the derivation for the parametric equations will mirror all our previous derivations and thus give the exact same result. This is a great example to see how we can spot patterns in the derivation of parametric equations that involve trig functions, so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuAu4o8BZJsHTf-FQg
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-curves-example-9-the-cycloid-proof-part-3
Related Videos:
Parametric Curves: Example 8: The Cycloid: Proof Part 2: https://youtu.be/_LAmOMbXXts
Parametric Curves: Example 7: The Cycloid: Proof Part 1: https://youtu.be/XmHpIZqeRKk
Parametric Curves: Example 6: Graphing Devices: https://youtu.be/cwpEasc9mhQ
Parametric Curves: Example 5: Lissajous Figure: https://youtu.be/0OT1OnnAQ9I
Parametric Curves: Example 4: https://youtu.be/WZYIS1My-g4
Parametric Curves: Example 3: https://youtu.be/VpiJdEviBbk
Parametric Curves: Example 2: https://youtu.be/SBkzZDlqTTE
Parametric Curves: Example 1: https://youtu.be/0k7QyWa9-HA
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Angles - Degrees vs Radians: What are Radians??: http://youtu.be/bOdrVvQWRdE
Trigonometry Identities: cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y) : http://youtu.be/VuQczhk7HOs
Trigonometry Identities: sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y): http://youtu.be/edtk9thfwbM .
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https://www.youtube.com/watch?v=4VuYJRFAI_g
In this video I derive the formula for the distance between a point to a plane in 3D coordinates. The distance formula can be derived by taking the scalar projection of a vector from a point on the plane to the desired point and projecting it onto the normal vector perpendicular to the plane. We can then utilize the geometric interpretation of the dot product to get a formula involving cosine and our desired distance. Further algebraic simplification yields the distance formula.
The timestamps of key parts of the video are listed below:
- Example 8: Distance from a Point to a Plane: 0:00
- Distance = Absolute Value of Scalar Projection: 1:15
- Distance Formula: 10:20
This video was taken from my earlier video listed below:
- Equations of Lines and Planes: https://youtu.be/qWQz6qPhXR8]
- Video notes: https://peakd.com/hive-128780/@mes/equations-of-lines-and-planes
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FO3u0IupqllNNkZTsffpIV
Related videos:
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https://www.youtube.com/watch?v=ZxSAc32uIqU
In this video I use the ideas and concepts from my earlier video where I introduced the Area Problem (see video link below) to derive an expression and hence a definition for the area under any continuous function.
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIiZ1IdLddAktuKrfNyA?e=jJ8ZB7
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/general-area-of-a-region-definition
Related Videos:
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What are Limits? A Simple Explanation: http://youtu.be/FbV7TzlkTZk
Sigma Notation - A brief Introduction: http://youtu.be/Gew7y73NY30 .
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In #911Truth Part 12 I have uploaded the infamous March 2, 2007 interview between Democracy Now’s Amy Goodman and Retired US General Wesley Clark regarding the United States PRE-PLANNED “7 countries in 5 years” invasion of the Middle East. Clark details how a mere 10 DAYS after 9/11, a general at the Pentagon told him that the decision to go war with Iraq was already made! And the reason given was “I don’t know”…. In other words, 10 days after 9/11, and before the media driven lie about “Iraq WMDs” the decision had already been made. Furthermore, Clark reveals that several weeks later, upon which the United States were already bombing Afghanistan, he went to see the unnamed general again to see if the Iraq War plan was still going to happen. But the General revealed that it was much worse and gave him a classified memo that listed “7 Countries in 5 Years” to be “take(n) out”: Iraq, Syria, Lebanon, Libya, Somalia, Sudan, and finally IRAN!
Now while the United States war machine has taken longer than “5 years” to carry out their plan, the first 6 countries have already been decimated resulting in mass genocide, a global immigration crisis, and the ballooning of fiat debt money across the globe that further increases the taxation upon ordinary hard working people. And even though Clark said he heard of the plan for the Middle East “War” aka Mass Ritual Sacrifice and Genocide "10 days after 9/11", it is clear the plan was made well in advance; and thus so too was the planned media brainwashing on a global scale planned as well.
There is now one country left on the list: IRAN. And from the current Puppet President, Donald Trump, and his “demented” tweets to the blatant government run Psychological Operation known as “Q Anon”, the level of outright threats towards Iran make it clear that the agenda for War will continue; but this time towards a more global war that includes Russia and China.
Since the war agenda continues regardless of who is president of ANY country, the inevitable realization comes to mind: we (as in the average ordinary citizen) are under global control. In fact, I would go so far as saying that the global power structure is not made up of “countries” but rather a two tier class: those that rule, and those that are ruled. But on an even deeper philosophical level, it is the very POSITION of power over others that will always lead to war; in other words, it doesn’t matter who is the president, even if it was yours truly, the result would be the same. Those that rule do so simply because they believe the global population are not fit to rule themselves, and everyday we prove them right by our very lifestyles, and ultimately by our very willingness to kill when asked to. Afterall, the “global elite” can only do so much in making us think that killing our fellow human being is EVER a viable decision. And ultimately the decision for peace, for self-governance, and saying no to war of ANY kind rests with us. #BoycottWar
Stay Tuned for #911Truth Part 13…
Full Amy Goodman and Wesley Clark Interview: https://youtu.be/JOtbNC4oJ54
“7 Countries in 5 Years” Transcript: https://genius.com/General-wesley-clark-seven-countries-in-five-years-annotated
View video notes on the Hive blockchain: https://peakd.com/war/@mes/911truth-part-12-gen-wesley-clark-reveals-middle-east-invasion-was-pre-planned-and-iran-is-next
Related Videos:
#911Truth video series: https://peakd.com/truth/@mes/911
#MESScience video series: https://mes.fm/science-playlist
#MESExperiments video series: https://peakd.com/mesexperiments/@mes/list
#AntiGravity video series: https://peakd.com/antigravity/@mes/series
#FreeEnergy video series: https://mes.fm/freeenergy-playlist .
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In this video I demonstrate that a mountain's shadow on the clouds can only occur when the Sun is near or below the horizon, thus proving the Earth is not flat. When the Sun goes below or near the horizon, it can create shadows of tall objects such as buildings or mountains that are visible in the clouds, creating some stunning imagery. This phenomenon can not occur on a Flat Earth since it would literally mean the Sun is at ground level. Some "Flat Earthers" try to deny the obvious by stating the Sun is somehow reflecting off the ground and then back on the clouds which is too ludicrous to even entertain. Nonetheless, I do a simple table top experiment showing that the shadow of the Mountain can only go above it's elevation if the light source was below the mountain. I also test out what happens when the light is shown onto a mirror on the ground, which results in some interesting but very different shadows than what are actually observed in nature. In later videos, I will debunk further bogus Flat Earth "explanations" by the likes of (disinfo agents) Eric Dubay and Taboo Conspiracy, so stay tuned!
Video links:
- MES Links: https://mes.fm/links
- Debunking Flat Earth Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUyOvHJiyX74GmJ79exCkh
- This video CREATES EX Flat Earthers by SeekTruth | SpeakTruth (STST): https://youtu.be/USt4uTbIOKk
- Shadow on Clouds: https://flatearth.ws/shadow-on-clouds
- Flat Earther says Sun is reflecting off the ground LOL ???https://youtu.be/6ocJ1Jo0ppU
- Mountain Shadow on Clouds Experiment: https://youtu.be/njvOTb1BJYg
- DRAFT Experiments: https://www.youtube.com/playlist?list=PLai3U8-WIK0FfZ_7hUuyO7xN8lp5oaDiu
Timestamps:
- Intro and Links: 0:00
- MES Links: 1:01
- Mountain shadow on clouds photos and information: 1:39
- Flat Earther bogus reflection "explanation": 3:24
- MES DRAFT experiment: Shining light on container: 3:38
- Shining light on mirror has different shadows: 4:28 .
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In this video I demonstrate the amazing drop down list function of Microsoft Excel and show you can also use it with tables to force users to select from a pre-determined list. This is extremely useful so make sure you understand how to create your own drop down lists!
Download the Excel Notes in my video: https://1drv.ms/x/s!As32ynv0LoaIibFH09RC0LEWWFTS_w?e=Tyl8rk
Related Videos:
Intro to Tables in Excel 2007: http://youtu.be/B_zXmunqrl4 .
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