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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.

$$y=\ln x, y=1, y=2, x=0 ; \quad \text { about the } y-axis$$

$$

\frac{\pi e^{2}}{2}\left(e^{2}-1\right)

$$

Applications of Integration

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Numerade Educator

Missouri State University

University of Michigan - Ann Arbor

Idaho State University

find value given the equations. Why equals Ellen of X? Why equals one? Y equals two x equals zero, and we're rotating around the Why access. So the graph for the natural log of X looks like a log rhythmic function and just gonna kind of estimate that at this point, obviously, there's not a whole lot of exact points on the graph of Ellen of X. But I do know that it happens at 10 Ellen up one gives us an output of zero. So I do know that one good ordered pair here and then the rest of it, like I said, kind of just that log shape. So then let's fill in our vertical and horizontal lines as well. So the horizontal lines we have our why equals one y equals two. So why it was one y equals two. Well, you know that's not horizontal better. And then the vertical line X equals zero. So if you look carefully, the area that's bounded by those four things is actually this area here between one and two between X equals zero and then the function Ellen of X. Now we're going around the y axis. So as I look at that shaded area to the y axis, there's no missing pieces, right? All comes up front right next, it right next to it. Which means that is a disc method. So for the disk method, your value is pi r squared in the in a girl and you use whatever variable your revolving around this case D Why? Because I'm going on the y axis, So I need by y one and y two, we already have our Y values based on the boundaries. 12 So then the only other thing we need to fill in is that radius that are squared is tell me the radius of the figure that's being revolved. The radius obviously goes from the center to the edge, and the center is happening wherever you're revolving. So here the Y axis is my center. The furthest out I go is to the graph of Ellen of X. So for the distance, for our I would think about what's my right minus left function, which would be Ellen of X minus zero, is the y axis. But look, if I feel that in what's wrong here, I don't have my variables the same. I can't integrate X with respect to why and expect to get the correct answer here. So I do have to change this to be in terms of why So I need to solve Why equals Ellen of X for X equals a function with respect to y. So to get X by itself, we need Teoh use e remember he rewrite the natural log. It's saying e toe what exponents gives me x e to the y gives me X So this is actually my function solved for X equals. And that's what I want to fill in here as my radius because you want to use the variable. Why so e to the y is what I wanna have here. This in a girl is a pretty easy you substitution. If you re right e to the y squared as each of the two y and then you can make to why you're you which means the derivative your do you is to de y so rewriting this integral would be 1/2 e to the you Do you so like I said, pretty easy to go through and solve it by hand here, obviously, if you have a calculator you can go through and just calculate it. Once you're at this step, but finishing out the problem, then your entire derivative of each of you is eat of the U and I'm gonna go ahead and change it back to wise. So I get 1/2 chi e to the fourth minus 1/2 pi e to the second filling in upper bound, minus lower bound. This is a totally acceptable answer, or you could factor out what they have in common. They have a 1/2 PYY squared in common, which doesnt really make it look any nicer. But it's a factored form of your answer. Both are equivalent. Both are acceptable for sure.

Oklahoma Baptist University

Applications of Integration