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Numerade Educator

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Problem 19 Medium Difficulty

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ x = 2y^2 $ , $ y \ge 0 $ , $ x = 2 $ ; about $ y = 2 $

Answer

$V=\frac{13 \pi}{3}$

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Video Transcript

We know that if X is to y squared than what we have is why is one again reuse the method of cylindrical shells, which means our our is to minus y and r h is gonna be to minus two. Why square, in other words, to minus explore using why cats do wise the two minus two. Why square? Which means plugging in what we know we have to pie times two minus y times two times one minus y squared. Now we need to look at this in terms of the integral from 0 to 1 to minus y times one minus y squared, Do you? Why, I would recommend using the power rule me integrate It makes a lot easier because now what we're doing is we're increasing the exponents by one. And then we are dividing by the new exponents just like us. Which means plugging in. We have four pi times one to the fourth, which is just one minus two times one cubed, which is just again one times two on the top. One squared over two plus two times one, which gives us four pi times what's on the inside. So we end up with 13 pie over three