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University of North Texas

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Problem 45 Hard Difficulty

Find the volume of the solid generated by revolving each region about the $y$-axis.
The region in the first quadrant bounded above by the parabola $y=x^{2},$ below by the $x$-axis, and on the right by the line $x=2$

Answer

8$\pi$

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Video Transcript

we want to find the volume of the solid generated by all being the region. Why is even X squared the X axis and ecstasy to around the why access. So this here is a sketch of that region and the first single gonna want to do is right out what equation we want to use. So since we're rotating around the why access here, this chapter says we should use the equation four interval from a to B of some outer radius, which I'll call Capital R. And we want this with respect. Why minus some Inter Radius Joe called Little are and I will also be with respect why? And we also need to multiply this by pi. So the first thing we're going to need to do is figure out what our radius is are going to be. So we're going to start from the live access and then just draw a little lying to the first thing we hit. So the first thing we hit is why is even two X squared? So one thing we're going to need to do with this is rewrite it so we can get it in terms of the function were actually interested in. So we want to rewrite this in terms of why So I'll go ahead and square root each side so end up with screwed of why is equal to X. So the distance from the ex access to this blue line should be the square root of why. So we can go ahead and write that our little why should be the square root of why and we'll do the same thing to figure out what our outer radius should be. And it just ends up being X is equal to. So are capital R is just going to be two now we can go in and blow these values and so high I'll leave all our bounds and integration is gonna be for right now. So our outer radius was too. So squaring that gives us or are in a radius should be the square root of y and squaring that will just give us why. And then we have our do I on the end here. So, looking at our graph, what we need to do is figure out what our range of why values will be so in this case, one of them is fairly obvious. It should just be Why is equal to zero since remember, we're also bound below by the ex Access, and then to find our upper bound will need to find where the Line X is equal to two. And why is he going to X squared Inter sex each other? And so all we need to do for this is take two and plug it in. So I'm gonna take that to plug it into their and that just tells me why is equal to four. So our upper bound for this will be four. Now we can go ahead and integrate so high, so integrating four would give us or why and then integrating. Why will these power rule? So now it's going to be, why squared and then we divide by the new power. And then we want to evaluate this from 0 to 4. So I'll go ahead and plug foreign. So four times four is 16 minus four square to 16 divided by two will be eight. Then we subtract off when we plug it zero. And so for why zero and zero squared zero. So the second term just become zero and the 16 minus eight is eight. So we end up with a pie. So the volume of this solid when we revolving around the why access will be a pie.