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Numerade Educator

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Problem 28 Hard Difficulty

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_3 $ about $ OC $

Answer

$\frac{2 \pi}{9}$

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Video Transcript

we're getting a figure with a region in the line and we have to find the volume generated by rotating this region about this line. The region is R. three and the rotating it about the line. zero. C. Sorry? Oh see who gets the origin. Mr. Now our region has top boundaries. Y equals the portrait of X. Bottom boundaries. Y equals X. Yeah. And zero or O. C. This is the same as rotating about the Y axis. So looking at our figure, we get a washer type solid. So we'll use the washer method. Now we need to find the outer and inner radius. So the outer racialist is equal to see X equals Y minus X equals zero. Which is it's X equals Y. Oh and the inner radius going on in Wisconsin. I worked for the clinton campaign anyway. The nazis. This feels already cold news. It sucks. The news sucks now after what X equals Y. To the fourth minus zero and therefore are volume V. Is equal to pi times the integral from Y equals zero. two. White equals one of the outer radius, Y squared minus the inner radius. Why the fourth squared. Dy. Mhm. And this is a simple integral to evaluate. Once you do, you should get two Pi over nine.