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Problem 61 Hard Difficulty

Find the volume of the described solid $ S $.
The solid $ S $ is bounded by circles that are perpendicular to the x-axis, intersect the x-axis, and have centers on the parabola $ y = \frac{1}{2} (1 - x^2) $, $ -1 \le x \le 1 $.

Answer

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

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Video Transcript

So for this problem, we first going to look at the region that we have, Um and it's going to be why equals one half, one minus X squared. And we're restricting the domain. I'm from negative one one. So the graph that we have on, we're going to, ah, rotate this eso we want to look at radius. And since they're rotating it around the y axis already, this is gonna be why So now what we have is that the equals Hi, I'm the integral from negative one toe, one of the radius squared the radius is gonna be Why so why squared the X and then we know why is eso we want to square Why? And we end up getting, um, hi over to we can take out that constant and then it's going to be on the inside one minus for squared class next to the fourth. We can take the integral of this. And what we end up getting is ultimately we can do is power to, by changing this to zero. So what will end up getting if you know, this correctly will be for pie over 15. Okay. Um and ultimately the best way to get this would be through evaluating the integral, um, well, have pi over two from 011 minus tease values and plus this value evaluated a d. X, and this would be our final answer for pi over 15.