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Problem 27 Hard Difficulty

Each integral represents the volume of a solid. Describe the solid.
$$ \text { (a) }\pi \int_{0}^{\pi / 2} \cos ^{2} x d x \quad \text { (b) } \pi \int_{0}^{1}\left(y^{4}-y^{8}\right) d y$$

Answer

This region rotates around the y-axis.

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Video Transcript

Let's consider the integral well, pie times the integral from zero to pi over two of co sign squared X dx. Okay. And the objective here is to describe the solid, um of that the integral represents the volume of the salad. Okay, so what we're gonna do is we're gonna use the result here that the integral from A to B of A of X, the X um represents the volume of the salad obtained by rotating the region bounded by the curve. Um, why is equal to f of X? Um so rotating the curve, um, rotating. So this integral, you know, it'll be of a of X. TX represents the volume of the solid obtained by rotating the region, bounded by the curve y equals F fax and bounded by the lines. Um, X is equal to a and X is equal to be all right. So what Bounded by Ryan's ex, is he gonna A and actually going to be. So the volume, um, is given by the volume V is equal. Chew well pi times the integral from zero to pi over two of co sign squared X dx. Okay, um which is equal to well, the integral from zero to pi over two. Um, the pichon come in Such This is pi Times co sign of X squared the X Okay on. And we can rewrite this as well. The integral from zero from X equals zero to exodus. Private two of a of X t x is equal to the integral from zero to pi over two of, well, pi times Why of X squared DX. Okay, now we compare. Um, these two equations and we see that Well, a a of X is equal to pi times co Sign of X squared. Okay, we have a disk, and this is equal to, well, pi times. Why Off X square. Okay. And now, since since a evac since a of X is equal to high times the radius times the radius squared. Right. So here we have the radius up. Steve Radius is equal to co sign of X and were bounded by the lines so and bounded by well, the lines X equals zero and X is equal to pi over two. So therefore, the salad, um obtained by rate is by rotating the region, the region here is well, zero less than or equal toe. Why? Which is less than or equal to co sign of X and, um zero less than or equal to x, which is less than or equal to pi over two. Okay, about the X axis. So here is our region a solid API, my rotating this region about the x axis about the x axis, um, and we can actually see. Or the region here. The region of integration is Well, um so his is the origin. Let's say here is, um, So, 1/4 pie, then we have 1/2 so 1/2 pie. Then we have 3/4 pie. Okay, we have pie. Oh, here. Um, if we look at the Atlantic was through, um, so 1/2 pi were coming. Basically. So hearing from zero that's a up here is one, um, this region Look in something. You, um, something like that. And we see that our region is given right here. All right, take care. Right

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