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Problem 32 Medium Difficulty

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = 0 $ , $ y = \cos^2 x $ , $ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $

(a) About the x-axis
(b) About $ y = 1 $

Answer

a) 3.70110
b) 6.16850

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Video Transcript

So for this problem were given some bounds for integration Y equals zero y equals cosine squared X. You know X is between negative pi over two and pi over two and we want to do it about the X axis. So what we end up getting As a result, if we have our graph here, the shaded region is going to look like this. And then again, that's going about the X axis. So it's going to give us this solid. Um, and when we since this is going to be a disk, we're going to use the disk formula, so it'll be volume pie. Um, our bounds are negative. Pi over 22 pi over two and then our radius in this case is cosine squared X. So then we square that and we have derivative or the integral. So when we take the inter row of this, our area is going to end up being two pi times zero to pi over two of the courtesan to the fourth X DX. We can use our calculator here and what we end up getting is that we have a volume of 3.70110 then for part B. What we want is to do it about, um y equals one. So as a result, we're gonna end up getting a washer. So to do this washer, we're gonna use the washer formula. Remember, the washer formula is pi times the bounds times the honor roll with bounds A and B are one squared minus R two squared d x So now what we end up having is our volume being equal to hi. Going from negative pi over two two pi over, Tim, Our first radius is one. So one squared minus one minus cosine squared, X squared DX. We can simplify this further, and what we end up getting is pie. Actually, we can do to pie good from zero two pi over two times of to cosign squared X minus co sign to the fourth X. Now you can differentiate this and put it in our calculator. What we end up getting is for our volume 6.16 850 And remember, these answers are going to be in units cubed since we're dealing with volume