Notice that the height of each rectangle is and the width of each rectangle is.

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1 Math 1410 Worksheet #40: Section 6.3 Name: In some cases, computing the volume of a solid of revolution with cross-sections can be difficult or even impossible. Is there another way to compute volumes when such an instance occurs? Of course! Starting with a region R between the graph of y = f(x), the x-axis, x = a, and x = b, we know that we can approximate the area of R with rectangles and create Riemann sums. Start by subdividing [a, b] into n subintervals of equal width x = (b a)/n, just as we have done in the past. On each subinterval [x i 1, x i ], construct an approximating rectangle where the top of the rectangle touches the midpoint of the subinterval. This means the sampling points are x i = Notice that the height of each rectangle is and the width of each rectangle is. Suppose that we rotate R around the y-axis to get a solid of revolution S. We can approximate the volume V of S by rotating the rectangles and adding up the volumes of the resulting shapes. If we focus on one rectangle, then rotating that rectangle creates a cylindrical shell, which is a hollowed-out cylinder.

2 For a general cylindrical shell of height h, let s say r 1 is the inner radius of the shell and r 2 is the outer radius. Then the volume of the shell (which we ll call V shell ) is the difference between the volume of the outer cylinder and the volume of the inner cylinder: V shell = πr 2 2h πr 2 1h = If we let r = r 1 + r 2 2 V shell = be the average radius, and r = r 2 r 1, then the volume of the shell is Back to the solid S, as mentioned earlier, if we rotate the approximating rectangles, then we obtain cylindrical shells. For the shell generated by the rectangle on the interval [x i 1, x i ], we have r = h = r = This means the volume of this shell is V shell = Adding up the volumes of all shells generated by the approximating rectangles, we get an approximation of the volume of the solid S: V If we use more shells of decreasing thickness and add their volumes, we should get a better approximation of the volume of S. Letting the number of shells go to infinity, we get the exact volume, which is represented as an integral:

3 Theorem 1. (Cylindrical Shell Method) The volume V of the solid obtained by rotating the region below y = f(x), a x b, about the y-axis is V = Problem 1. Use cylindrical shells to find the volume of the solid obtained by rotating the region under y = x 2, 0 x 3, about the y-axis.

4 Suppose we instead rotate a region bounded by two curves y = f(x) and y = g(x), where f(x) g(x) on [a, b], about the y-axis. The formula for cylindrical shells is then: V = Problem 2. Use cylindrical shells to find the volume of the solid obtained by rotating the region between y = x 2 and y = x 5 about the y-axis.

5 If instead we rotate about the x-axis, then we shift our perspective by looking at functions of y instead of x. Let s say we rotate the region between x = f(y), the y-axis, y = c, and y = d about the x-axis. Then we have the same volume formula as before, but now the integral is in terms of y: V = Problem 3. Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region between x = 12(y 2 y 3 ) and the y-axis.

6 Similarly, consider the region bounded by two curves x = f(y) and x = g(y), where f(y) g(y) on the y-interval [c, d]. If we rotate the region about the x-axis, then we get a similar formula for volume as before, but again in terms of y: V = Problem 4. Use cylindrical shells to find the volume of the solid obtained by rotating the region between y = x, the x-axis, and the line x = 4 about the x-axis.

7 What if we rotate a region around a horizontal or vertical axis that is not a coordinate axis? How do we use cylindrical shells in this case? Let s consider an example to illustrate this situation. Problem 5. Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y = x 2 and y = x 5 about the line x = 2.

8 Just as in the last section, we will not consider volumes obtained when rotating around any line other that is neither horizontal nor vertical. Now that we have two different types of formulas for finding volumes (cross-sections and shells), which one do you try first? That depends on the situation. Let s put everything together in a table to help summarize the different situations: