43–55. Volumes of solids Choose the general slicing method, th...

Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?

Accepted Answer

Welcome back, everyone. In this problem, let R be the planar region enclosed by the lines Y equals 0, Y equals X and Y equals 4 minus X. Compute the volume of the solid obtained by rotating R above the horizontal line Y equals -1. Now how can we figure out the volume of our solid by rotating R? What method could we use? Well, we could use the washer integral for rotating about this horizontal line with a space in between, because recall that by the washer method. OK. It basically tells us that our volume is going to be equal to pi multiplied by the integral between the bones of A and B. Of our outer radius squared. As a function of X minus or inner radius squared as a function of X or with respect to x. So if we can figure out an expression for our inner radius or outer radius and the bones of our um region, OK, then we should be able to find the volume. So let's do that. First, let's find our bones. No bones will be at the intersection points of the lines Y equals X, Y and Y equals 4 minus X, and 4 Y equals 0, OK, which is on our graph here. No, for the intersection. So let me see here at the intersection. Of Y equals X and Y equals 4 minus X, OK. Then X will be equal to 4 minus X, which means that 2 X will be equal to 4, and thus X will be equal to 2. And if X equals 2, that means Y equals 4 minus 2 or 2, OK? So, at our point of intersection, which we can also see from the graph, is 22. Then that tells us that all three lines meet at our origin 00. The 0.22 and the 04 0. OK. So, and we can see that region here are the triangle with the base and the X axis for X going from 0 to 4. Now remember, we're told that the axis of revolution is the horizontal line Y equals -1, because this is the line about which the region is revolved. So we can use that to help us figure out the radii for our washers. Now for our inner radius, OK, the inner radius. The inner radius are of X, OK. For our inner radius. Then it's going to be the distance from Y equals 0 to Y equals -1. So R of X, the inner radius equals 1. On the other hand, for the outer radius. Which is capital R of X, OK? Then it's going to be the distance from the top of the region to Y equals -1. I know you notice that that distance differs, OK? For example, there's a different distance from the top. Up to uh from the top to Y equals -1 for Y equals X and for Y equals 4 minus X down to Y equals -1. So all of that depends on which region we are in. So if I were to highlight that here, the alter radii changes for 0 to 2 and from 2 to 4. So generally it's equal to the top. X + 1, OK, where it would be equal to. X minus X + 1 because Y equals X for the top X in the interval from 0 to 2. And it would be equal to 4 minus X + 1 or 5 minus X on the interval from 2 to 4, OK? So, now that we have both of these values, that means we'll have to integrate separately depending on whichever auto radis we are using. In other words, we're going to split our interval at X equals 2. So, know that we have our outer radii, check our inner radius, check, and our bones, OK, from going from 0 to 40 to 2, and then from 2 to 4, OK. Then, this tells us That by splitting our volume, it's going to be equal to. I multiplied by the integral between 0 and 2 of or auto radius X + 1 squared, OK. Minus or inner radius 1 squared with respect to x. Plus pi multiplied by the integral from 2 to 4, OK, of or outer radius in this case 5 minus X2 minus 1 squad or inner radius with respect to X. So now we can evaluate both of these integrals. Now here for our first term, OK, X + 1 squared equals X2 + 2 X + 1 and X2 + 2 X + 1 minus 1 would simply be equal to X2 + 2 X. OK? So we're integrating that with respect to X between 0 and 2. On the other hand, For our second integral, OK. Then 5 minus X2 equals 25 minus 10 X plus X2, and if we subtract 1 squared from that, 25 minus 1 equals 24. So this would be X2 minus 10 X plus 24. All with respect the X. Now if we integrate both of those, the integral of X squared is a third of X cubed. OK, and the integral of 2 X is Xsquared, so we're evaluating that between 0 and 2. And For other term, X2, the integral of X2 is a third of X cube. That's gonna be minus 5 X2 + 24 X. OK? That is between 2 and 4. Now for our first term, if we substitute 0 here, the second terms will disappear. So this will simply be pi multiplied by 203, OK? Because it would be 8/3 + 4. And on the other hand, if we evaluate our expression here between 2 and 4, we will also end up with 20/3s. OK, so that's also pi multiplied by 203. And 20/3 of pie plus 2/3 of pie would be equal to 40/3 of pie. Therefore, That means our volume when the region is R is rotated above the horizontal line Y equals -1 will be 40/3 of pi cubic units. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Volume of Solids of Revolution

Disk/Washer and Shell Methods

Setting up the Integral with Correct Bounds and Axis

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