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Problem 30 Hard Difficulty

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

$ \Re_3 $ about $ BC $

Answer

$\frac{4 \pi}{15}$

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Video Transcript

we know that in order to find the volume, we must integrate along the axis parallel to the axis of rotation. The axis of rotation, given the figure is wife was one. Therefore, the outer radius is gonna be one minus X because it's the distance from Michael's ex to the actors of rotation and the inner radius is going to be one minus X to the 1/4 is the distance from y equals the four through of acts or ex, the 1/4 to the ass of rotation, which we already established His y equals one. Therefore, now we're gonna be plugging into the formula the integral from A to B pi times outer radius squared mice in a radius squared so outer one minus ax squared, minus inner squared. I would recommend simplifying this before you plug in to the integral. No, we're gonna be integrating by using the power rule, which means we increase the exponents by one and then divide by the new exponents. Plug in. We know for the bottom bound, we simply have zero, which doesn't affect the solution. Consult it like charms. And we have four pi over 15