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Problem 13 Medium Difficulty

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ x = 1 + (y - 2)^2 $ , $ x = 2 $

Answer

$\frac{16}{3} \pi$

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Video Transcript

using the method of cylindrical shells. We know that we have to set equal toe one post y minus two squared because we have to find the intersection points for why those are gonna be our to bounce. So we have to do this. So? So for why, again? We're trying to get why equals, once we have factored, we know we end up with our wise one and three. Good. Those are two bound. Now we have this. We know that if ours why and the height is gonna be two minus one plus y minus two squared. Then we need to simplify the height also to make it easier to plug. And otherwise it's gonna be really messy. This is negative. Y squared. Plus four. Why last three? Ok, good. That's something we can now put into the integral. Now, remember, the formula for interval is to pied times bounds, which in this case is from 1 to 3 of two pi r h do you want are two pilots That is a Constance. We already put it out there. So R h d y wide times negative y squared plus four. Why minus three d y okay Now, when we integrate, we use the power method, which means we increase the expert at 51 and then we divide by the new Expo in This is the easiest way to integrate. We know we have our bounds from 1 to 3, which means now we can actually plug in our batons. We have two of them, which means we have to do top minus bottom, and near of them are zero, which means we actually should plug in the bounds. Oftentimes, when we have zero is one of the bounds, it ends up being that zero plugged in. Just get zero so weakened neglected. However, in this case, we can't because we have one on the board. I'm not zero, which means we actually have to plug it in and see what we get. We end up with two pi times four minus 4/3 which gives us 16 divided by three pie. When you do the algebra