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University of North Texas

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Problem 53 Easy Difficulty

The region between the curve $y=1 / x^{2}$ and the $x$ -axis from $x=1 / 2$ to $x=2$ is revolved about the $y$ -axis to generate a solid. Find the volume of the solid.

Answer

$4 \pi ln(2)$

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Video Transcript

we want to find the volume of rotating the region between the curves. Why is he gonna one over x squared the ex access as well as acceptable to 1/2 an ex secret to which I went ahead and graft this region. So the area we're interested in is this right here and we want to rotate this area around the ex access. I mean, the why access. So we just come out and rotate something like this for the 1st 1 and then we would do the same thing for the other ones. So it looks like we're drawing cylindrical shells. So to find the volume for the shell method, this should be to pie integral from a to B oh x times f of x dx. So now we can go ahead and plug everything in, Um, which would be let's see, So you have to pie A is 1/2 B is too. So I have one have to too, and then x times f of X, which in this case is one of the X squared DX. Now that simplifies to one over X. And we learned from this chapter that that would be the natural log of the absolute value of X. And we evaluate from 1/2 22 All right, now we can go ahead and plug in to plug in 1/2. We have two pi natural log of two minus the natural log. Oh, 1/2. And pulling this negative in would make that natural walk up to so we'd have natural log of two plus natural look up to which is going to be four pi natural log of two for our volume.