In this video we are going to learn how to use integration to calculate the center of mass of a region. The idea here is that we will have a function y = f(x) and we will look at the region underneath it from x = a to x = b. We are going to imagine that we cut out a thin plate that fills that region. We want to figure out exactly where the center of mass is. In other words, if we were going to balance this region on a particular point, where would it balance. The center of mass is also known as the centroid and in some of the examples this word will be used.
formulas are not symmetric. You dont get from one to the
other just by switching xs and ys!
you will have to do another integral to find the area,
However, sometimes you can find it just using geometry, such as the
standard formulas for areas of rectangles, triangles, and circles.
the region has horizontal or vertical symmetry, you can use that to
find the corresponding coordinate of the center of mass without
its feasible, check that your answers make sense. Graph the
region and locate the coordinates of the centroid that you found.
Are they inside the region? Are they approximately where it looks
like a plate of that shape would balance?
centroid of a triangle with a horizontal base is always 1/3 of the
way up from the base.
Center of Mass
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