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Problem 8 Medium Difficulty

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.
$$y=\frac{1}{4} x^{2}, x=2, y=0 ; \quad \text { about the } y-axis$$

Answer

2$\pi$

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Video Transcript

you if I have the equations. Wyck was 1/4 X squared X equals two y equals zero, and we're going around the why access to first ball 1/4 x squared is a parabola, and it's like a shortened proble. X equals two would be a vertical line over where access to and why why equals zero is the X axis. So the area that has been bounded by those equations is this part of just colored in with that mind. If I'm rotating around the Y axis, then I do see that there's this missing space. As I rotate that top piece look at all this distance. It has to cover where there's nothing filled in. So that means this is a washer set up and for the washer set up, volume is pi times the integral capital R squared minus little R squared. You want to use whatever variable is your excess or parallel to your access. So in this case, D Y for the Y axis, and that's also gonna tell me my bounds. Why wanted Why, too? The bounds for this area are why equals zero that's given as that first equation, but for the 2nd 1 up here for why, too? It's like, what is this value? So we do know something about where that point happens. That point happens when X is too. So if I can use my equation for why knowing that X is too, then I can find that y value. So why you ghouls? 1/4 2 squared. Well, that's 1/4 of four. So my upper bound is gonna be one. So we're going from 0 to 1. We have capital R squared, minus little r squared. So now we need to consider what is capital art? That would be the radius if this whole thing were filled in and actually notice If you look at the bottom of the shaded area, that is the radius, if everything were fill in there is that one relief in infinite strip there that goes all the way from the access to the edge of our shape. And that's defined by the value of two. That vertical line back to the axis zero. So for capital are I'm gonna fill in two and then for lower case R. It's the stuff that I need to take away. So it's the stuff here in the middle that's missing notice. The stuff that's missing goes from the parabola back to the access. It goes from that curved shape back to the axis. Well, I can't fill that in because if I feel that in, I have the wrong variable. So I need to Saul to be in terms of Why. So let's solve this 1/4 ex squared to be in terms of why we're gonna have to multiply before over and then square root both sides. And I don't want the plus or minus here because obviously the exes, only on the right side of the origin it's only the positive version. So I don't need that plus or minus. So two squared of why is what I want to fill in here. So at this point, I'm ready to go ahead and solve this integral. I have four minus four. Why, if I square that second term so as I integrate, I get pi times the whole thing for why minus two y squared going from 0 to 1, filling in the upper bound of wine is four minus two You and then, if I feel in that lower bound of zero, it's nothing. So the simplified answer here is to buy for my volume