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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}{x} $ , $ y = 0 $ , $ x = 1 $ , $ x = 4 $ ; about the x-axis

$V=\frac{3 \pi}{4}$

Applications of Integration

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Missouri State University

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

and this problem. We are talking about a direct application of integration and that is finding the volume under a curve. So how do we do this? Um, there are two methods. There's a cylindrical shell method and washer method. We're going to be using the washer method. And so this will make more sense when you talk about the function were given. So this star coordinate plain and this blue line is our function Thes dotted lines are the interval that we're investigating and this red line is essentially the height of our function. So now what we do is we essentially rotate this or reflected across the X axis. So you can imagine we have this really long line here and we essentially call it a disk washer and we essentially add up all of these washers together to find the volume that would be created by our function. So you can see that this is a direct application of integration. We're finding essentially a sum of all of these washers, which is what an integral will do. So now let's find the volume for the function were given. So the volume is going to be equal to pi times the integral from 1 to 4 of our radius times dx the radius again is this red line. Imagine that this would be a full circle. This would be the radius. So now we could plug in some information. Our volume would be equal to pi times the integral from 1 to 4 of one over X squared in d X then we can. And to derive this, our volume would be equal to pi times Negative one over X evaluated from 1 to 4. So our volume is pi times negative 1/4 plus 1/1 and we can simplify that to three pi over four. So what we just found is that using the washer method and the function were given the volume that would be created by that washer is through pi over four. So I hope this problem helped you understand a little bit more about how we can use the integral to find the volume under our function. Even specifically, how we go about, um, one of these problems using integration and a given function

University of Denver

Applications of Integration