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 about the vertical line X equals -1. Now if we're going to figure out the volume of the solid obtained by rotating R, since we're rotating about the vertical line X equals -1, we could use the cylindrical shell method with vertical slices and integrate in X. Recall that by the shell method, yeah. We know that volume V is going to be equal to 2 pi. Multiplied by the integral between the bones A and B of the radius in terms of X multiplied by the height in terms of X with respect to X. So if we can figure out the radius, the height, and our bonds A and B, then we can integrate and solve to find the volume of the solid obtained by rotating R. So first, let's start by identifying our region clearly, OK? So we have our graph and we know where R is and notice that we have 3 endpoints on R. 3 points between which our region is enclosed. Now, we know that 00 is one of those points at the origin where Y equals X and Y equals 0 meet. And we also know that 40 is the point at which Y equals 0 and Y equals 4 minus X meets. So what about the point where Y equals X and Y equals 4 minus X meets? Well, we could use the graph, but we could also confirm it doing some a bit of algebra here. So notice that at the intersection. Of Y equals X. And Y equals 4 minus X. Then X will be equal to 4 minus X, which means if we add X to both sides, 2 X equals 4, and thus X equals 2. So both of these points will intersect at X equals 2 and where Y would be equal to 4 minus 2 or 2. So they both intersect at 22, which if we look on our graph visually, that makes sense. That adds up, OK? So the lines are, the lines meet at 0040 and 22, where the region is a triangle with the base on the X axis for X in the interval from 0 to 4. Now that we found A and B, what about our radius and our height? Well, notice here that the radius. The radius and. Is going to be the distance to the axis X equals -1, so it's equal to X + 1. No, for our height. OK, the height depends on the top, the, the, the top value of X or the equation with the highest value of X or the value of X that applies to the section. So for example, here our height would be equal to the top X minus 0, which would be the top value of X, but depending on our interval like we said, that changes. So notice that from 0 to 2. The top value of X would be based on Y equals X, while from 2 to 4, the top value of X would be based on Y equals 4 minus X. So. We could say then that. The first value of the first top value of X, let me put that here. Let me make some space. The first value for the top value of X would be equal to X, where X is in the interval from 0 to 2, while for the second value for the top value of X, it would be equal to 4 minus X when it is in the interval from 2 to 4. So now that we have all of these, we have our heights for our different intervals and our uh and the radius and the bones. Then we can find the volume because now we'll have to split our integral at the point where X equals 2. So really it's going to be 2 pi multiplied by first the integral between 0 and 2 of the radius X + 1 multiplied by the top X on that interval X with respect to X. Next, we're going to add the interval, the integral from 2 to 4 of X + 1 multiplied by 4 minus X, the top X between 2 and 4 with respect to X. So let's now see if we can go ahead and solve. Now if we expand our integral here, OK, this is going to first be the integral between 0 and 2 of X2 + X, OK, with respect to X, and for the second one, between 2 and 4 it's going to be the integral of negative X2 + 3 X + 4. With respect to X. And now if we integrate both. For the first one, OK, it's going to be the integral of X2, which is a third of X cubed, plus the integral of X, 1/2 of X2, OK, between the bones of 0 and 2. And now we're going to add that to negative X cubed divided by 3, OK, plus 3 halves of X squared, plus 4 X. Between the bones of 2 and 4. Now here for our first expression, we know that the lower bound is 0 and that will make our term 0. So the first one will be 14/3. And for a second term, when we worked up between the bones of 2 and 4, we should end up with 22/3. I know 14/3 plus 22/3. Would be equal to 36/3 and 36/3 multiplied by 2 i would be equal to 2 multiplied by 12 or 24. Therefore, the volume of our region would be 24 cubic units. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Volume of Solids of Revolution

Disk/Washer Method

Shell Method

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