Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = y^2 $ , $ x = 1 - y^2 $ ; about $ x = 3 $

$\frac{10}{3} \sqrt{2} \pi$

Applications of Integration

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Numerade Educator

Missouri State University

University of Nottingham

Idaho State University

for this question, we're finding the volume of a solid obtained by rotating the region bounded by X equals y squared. Thanks. People's one minus y scored rotating about the X equals three line. So here's our axes. You can be in with the X equals Y squared, which looks like a problem opening up to the right. Right? And then, um, the X equals negative. Y square typically looks like something out of this, but we're adding one. So we shifted to the right one. There's the exit polls, one mine. And so that's what the problem looks like. We have rotating about X equals three line. So that's right there. Yeah, basically rotating this region right here. This is getting us washers. Now, if we look closely at the region, we noticed that is symmetrical so we can divide into two that makes her integral easier. And then we can multiply the answer by too, in the end to find the total body. Now, when you find the points of intersection specifically the Y points here and right here. So that way we know where did integrate over. So the way we do that is set the two curves equal to each other. So we have X equals y squaring the Y squared equals the other one is one minus five squared moved the Y scores to one side. Divide by two. This is equal to square to over too. So now we know that square every two over to that's negative squared to over, too now. Before I said we could split it up into two sections. So now we can just integrate from zero to get you over to. Then once we integrate that, we can multiply the answer by to to find that find to find along. So now let's look for the cross sectional area. It's going to be a function of why, and it's done favor Ariel. Mine is a smaller area to bigger area. Being a disc is caused by this Kurt, which is the X equals y squared G'kar. So to find the distance between knackered in the X equals three line cigar three minus exactly, which is why I squared. This curve is creating the smaller radius. So to find the distance from Hacker to X equals three line, it's three minus one minus y squared, and so we can plug that in. We'LL call it the pie. Now the only way we can continues to simplify this whole out we're multiply it all out. Now we can combined, like terms in Super Five and take out the side. So that is our final prosection, period. Now we can go ahead with the girl. So we're integrating from zero to two from To are rather just brute you over too, And I'm gonna pull it the pie in the fire. Remember that when we're done with all this, we're gonna multiply the entire thing by two again because we're only doing half the bottle. Now it unplug it. One minus two squared in and about the anti directives, That's why. Get two thirds. Why did three, two, right? Two over two. Now we can just plug in himself. So first plug in there, too. And when we plugged in, Weikel zero, we just get cereal. So we don't really care about that now. Just cube the ritual to modify by two thirds. All right down here. Continue to simplify. I come Denominator. Wait. It's a silencer. Is ten two over three