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Harmonograph in action - YouTube - 0 views
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A harmonograph uses two pendulums swinging in perpendicular planes to create a graphic representation of harmonies. This device uses the same relationships found in musical harmonies, but at the much lower frequencies of pendulums. The design for this harmonograph came from "100 Amazing Award Winning Science Fair Projects" by Glen Vecchione, with minor alterations during the build and test phases. Another interesting book on this topic is Anthony Ashton's "Harmonograph: A Visual Guide to the Mathematics of Music".
Calculus Website by Kiryl Tsishchanka - 0 views
NYU > A & S > Academic Integrity - 0 views
Integration by Substitution by The Khan Academy - 2 views
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Here is another good video by Khan Academy explaining integration by substitution for those of you who may need help. There are also other videos on pretty much any topic in calculus which I find useful for reviewing and practicing. Khan Academy is on a mission to provide a free world-class education to anyone anywhere. With over 2,600 videos covering everything from arithmetic to physics, finance, and history and 200 practice exercises, we're helping students learn whatever they want, whenever they want, at their own pace.
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Students have a lot of problems with changing the limits when doing u-substitution with definite integrals. But it's easy to remember: if the variable integrated over is u, the limits have to be values of u. You start with limits which are values of x. How do you convert x to u? We have already decided that u=cos(x). So the limits x=π/2 and x=π become cos(π/2) = 0 and cos(π) = -1. He's being a bit sloppy by saying this computes a negative area. Area is always positive. The integral *results* in a negative number because the region we are finding the area of is below the x-axis. So the integral computes the negative of the area.
Lesson 27: Integration by Substitution (slides) - 2 views
Lecture 26 - Integration by Substitution - 1 views
Visual Calculus - Substitution - 3 views
Riemann Sum Applet - 2 views
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