The volume of the solid is. This is the differential element. This requires substitution. Thus we have:. With the Shell Method, nothing special needs to be accounted for to compute the volume of a solid that has a hole in the middle, as demonstrated next. In part b of the figure, we see the shell traced out by the differential element, and in part c the whole solid is shown.
In part b of the figure the shell formed by the differential element is drawn, and the solid is sketched in c. Note that the triangular region looks "short and wide" here, whereas in the previous example the same region looked "tall and narrow.
At the beginning of this section it was stated that "it is good to have options. Thus the volume of the solid is. This requires Integration By Parts. Yes, delete my work. Keep the old version. Delete my work and update to the new version. Cancel OK. How to use Ximera This course is built in Ximera. How is my work scored? We explain how your work is scored. A review of differentiation We review differentiation and integration. A review of integration We review differentiation and integration.
A review of integration techniques We review common techniques to compute indefinite and definite integrals. Accumulated cross-sections. What is a solid of revolution? We define a solid of revolution and discuss how to find the volume of one in two different ways. Comparing washer and shell method We compare and contrast the washer and shell method.
Applications of integration. Integration by parts We learn a new technique, called integration by parts, to help find antiderivatives of certain types of products by reexamining the product rule for differentiation. Trigonometric integrals We can use substitution and trigonometric identities to find antiderivatives of certain types of trigonometric functions. Trigonometric substitution. Trigonometric substitution We integrate by substitution with the appropriate trigonometric function.
Rational functions We discuss an approach that allows us to integrate rational functions. Improper Integrals We can use limits to integrate functions on unbounded domains or functions with unbounded range. Sequences We investigate sequences. Representing sequences visually We can graph the terms of a sequence and find functions of a real variable that coincide with sequences on their common domains.
Limits of sequences There are two ways to establish whether a sequence has a limit. What is a series A series is an infinite sum of the terms of sequence. Special Series We discuss convergence results for geometric series and telescoping series. The divergence test. The divergence test If an infinite sum converges, then its terms must tend to zero.
The Integral test. The integral test Certain infinite series can be studied using improper integrals. The alternating series test Alternating series are series whose terms alternate in sign between positive and negative.
There is a powerful convergence test for alternating series. Dig-In: Estimating Series We learn how to estimate the value of a series. Remainders for Geometric and Telescoping Series For a convergent geometric series or telescoping series, we can find the exact error made when approximating the infinite series using the sequence of partial sums.
Remainders for alternating series There is a nice result for approximating the remainder of convergent alternating series. Remainders and the Integral Test There is a nice result for approximating the remainder for series that converge by the integral test. Solution This is the region used to introduce the Shell Method in Figure 6. With the Shell Method, nothing special needs to be accounted for to compute the volume of a solid that has a hole in the middle, as demonstrated next.
Solution The region is sketched in Figure 6. In part b of the figure, we see a sample shell, and in part c the whole solid is shown. When revolving a region around a horizontal axis, we must consider the radius and height functions in terms of y , not x.
Find the volume of the solid formed by rotating the region given in Example 6. In part b of the figure the sample shell is drawn, and the solid is sketched in c. The following example shows how there are times when it does not matter which method you choose to evaluate the volume of a solid. In Example 6. We will now demonstrate how to find the volume with the shell method.
Note that your answer should be the same whichever method you choose. Solution Since our shells are parallel to the axis of rotation, we must consider the radius and height functions in terms of y.
Solution The region, a sample shell, and the resulting solid are shown in Figure 6. We must evaluate two integrals as we have two different sample slices. The volume can be computed as. While this integral is not impossible to solve, using the Shell Method gave us a significantly easier way to compute the volume.
The region, a sample shell, and the resulting solid are shown in Figure 6. The volume is. With the washer method, we need to integrate with respect to y because we are rotating around a vertical axis. We also need to divide the region in two because the washers will run into different boundaries at different heights.
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