Problem

For the following exercises, use a calculator to …

Problem 272 Medium Difficulty

For the following exercises, compute the center of mass $(\overline{x}, \overline{y}) .$ Use symmetry to help locate the center of mass whenever possible.
$\rho=2$ for the region bounded by $y=\cos (x)$$y=-\cos (x), \quad x=-\frac{\pi}{2},$ and $x=\frac{\pi}{2}$

Answer

since the figure is symmetric about $x$ axis and $y$ axis, it's center of mass is $(0,0) .$

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Calculus 1 / AB

Calculus 2 / BC

Calculus

Chapter 2

Applications of Integration

Section 6

Moments and Centers of Mass

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Video Transcript

in the following question. We want to find the center of mass located in the intersection between the functions. Why? Sickle to cosign effects on bicycle to native Kocian effects for X in between negative pirate chewed to pirate tube. We're getting that The density in between this reed in between these two functions is equal to two. So it's first pro on a proxy in a sketch of her intersection. Are the region are, which is the intersection of these dysfunctions? Looks like So we're here. We'll have the point. Negative one y school too negative one and one and then point pie Sequel to two Hi, Divided by two sorry and negative piety Right about you. So we know that cosign effects starts from one and then a pirate to zero. Then it repeats itself and it's the same around the negative. That was Fricks. So this is the line. Why sickle to co sign effects with green. And when I draw the line, why sickle to native coast and effects, which is the same. However, it's just reflected across the Y axis, so this will be the graph of negative ghost and effects. So this is why sickle to negative coast and effects. And this will be the region. Are Did you notice something similar in our well are has symmetry. Sorry around the X axis. And why does bar has symmetry around the X axis? Well, we know that this shape right here is the same once this one And because of that, we can see that it has similar around the XX is however eat old soon has symmetry around the y axis. And we know that because this shape right here and this one are the same. So because our has symmetry sorry, because this two functions have symmetry around the x and Y axes. We can actually compute the center of mass without actually using any integral. And in this case, is gonna be the intersection of this juice lines of symmetry which in this case is just one on this one, and they intersect at the 10.0 comma cereal. So because our because these two functions have symmetry around the x and Y axes, the center of mass is located at the intersection of these two alliance which

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