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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 know that in order to find the volume, we have to integrate along the axis parallel to the axis of rotation. Axis of rotation is X equals zero. There for outer radius is why the distance from X equals wide to the axis of rotation. An inter radius is why did the fourth the distance from X equals right of the 4th 2 axis of rotation, which we established over here is X equals zero. Therefore, V is pi times the integral from 01 of why squared, which is the outer minus. Why did the fourth, which is squared climbs do y? Let's simplify this before we integrate. Why squared minus one of the eighth? Okay, now we're gonna be integrating, which means we're gonna be using the power rule, which means that we increase the expert by one, and then we divide by the new exponents. Now we're at the point we can plug in our bounds, but yet zero for the bottom one, which gives us a solution of two pi over nine