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.
$ y = x^2 $ , $ x = y^2 $ ; about $ y = 1 $
Applications of Integration
for this question. We're funding the volume obtained by protein. They're region bounded by I equals X squared. X equals Y squared around the y equals one line so we can start off by drawing the South. So here the axes the y equals expert is pretty simple. It's just a problem opening upwards and then the X equals y squared is a problem opening towards the positive X axis. It's the exact same shape Rotated Now, Now, when you find so the region is this part that was written in red is enclosed by both cars. Um, the y equals one line could be drawing very easily as well. Another reason why I know at this point right here is one one. If you plug in and set them equal to each other, then the only two solutions are possible zero zero, which is right here in one one right here. So those are the two points of intersection, so therefore the y equals one line goes straight through that point of intersection. So we're rotating about that line. So hey was like that. And when we do that, we create a washer. So this inside creates a small disk in this outside creatures. Bigger disc. We're getting a washer of this with not with is changing as we increase X. So now when you find the cross sectional area and as a side before because of with changes with axe, the cross sectional area should also change with X cross sectional area of a disc where a washer is just pyre squared, his power squared. This is the bigger one. This is the smaller one. So let's find our one first. Our one is the big Aries, so obviously this point right here this curve creates the R, and this term is just the y equals X squared. So the distance between this point when the curve and the white was one line is just wine minus the Y value so equals one minus. The Y value is X squared, so that's our one. Our two is just discovered, and we know that it's the X equals y squared X equals Y square. We can express in terms of why that means why equals square root of X, and so the distance between this line and that a point on that curve is just one minus the y values well, which is square attacks. Now we know that, and we can plug that hole in so we can find the cross section area function. I'm gonna pull out the pie because pie is expressed in both terms. I'd even see her here. I'm seeing Prentiss. He's around back. It's not even light out some falling out this first term right here. And now it's time to find out this turn. I'm gonna do it within the privacy's. So when this court is one minus, two times go back plus squared of X squared is just that. So now we have to subtract everything in that. So one minus one is zero. So those two already cancel out. So that is our No, I mean a mistake. I forgot to subtract the entire thing. So this subtract minds plus Plus, it's just chatting. So this is our final cross sectional area function, and we can go ahead and plug it into the inner interval in a girl. So this interval is going from zero to one. What is this interval right here? And we can just plug in the cross sectional area function once again, I'm gonna pull out Pai to the very beginning, so this is a very long in a role, but it's very simple. Salt slightly tedious if they find the anti to remove of each of these terms. So first I have to tie in the beginning. The anti drug abuse, uh, negative to F Square is just keep the negative, too X to X squared. So this powers to you out of one his three. So you have to divide by three in an industry plus X four plus morning's five, and you also have to divide by five. That square of acts is just acts to one half right, so you can treat it as one have one half plus one is three house and you. When you divide by three halves, it's equal to san, um, multiplying by two third, which is it's reciprocal. So two times two thirds is four thirds Ben Teo untied your unjust acts this X squared. Not too. All this is evaluated at zero one. We don't really care about zero case. Suppose it all. Just leave that zero. So what we get is when we plug it in one negative two thirds. This one fits plus forthe church minus one have. That's it. No one would do this entire faction out. You get the final answer, which is Hello. Over thirty. So that is our final answer.