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Home Applications of the Integral Volumes of Solids of Revolution Cylindrica Shell Method  
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Cylindrical Shell Method
CYLINDRICAL SHELL METHOD: For rotations about the axis of the dependent variable. Figure 6.2.9 Let us again consider the region R under a curve y = f(x) from x = a to x = b, so that x is still the independent variable. This time rotate R about the yaxis to generate a solid of revolution S (Figure 6.2.9). VOLUME BY CYLINDRICAL SHELL METHOD Figure 6.2.10 Cylindrical Shell Method Let us justify this formula. Divide R into vertical strips of infinitesimal width Δx as shown in Figure 6.2.10. When a vertical strip is rotated about the yaxis it generates a cylindrical shell of thickness Δx and volume ΔV. This cylindrical shell is the difference between an outer cylinder of radius x + Δx and an inner cylinder of radius Δx. Both cylinders have height infinitely close to f(x). Thus compared to Δx, ΔV ≈ outer cylinder  inner cylinder ≈ whence ΔV ≈ 2πx f(x) Δx (compared to Δx). By the Infinite Sum Theorem,
Figure 6.2.12 (a) Figure 6.2.12 (b) Now let R be the region between the curves y = f(x) and y = g(x) for a ≤ x ≤ b, and generate the solid S by rotating R about the yaxis. The volume of S can be found by subtracting the volume of the solid S_{1} generated by the region under y = f(x) from the volume of the solid S_{2} generated by the region under y = g(x) (Figure 6.2.12). The formula for the volume is Figure 6.2.12 (c) Combining into one integral, we get VOLUME BY CYLINDRICAL SHELL METHOD Figure 6.2.12 (d)


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