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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 y-axis to generate a solid of revolution S (Figure 6.2.9).

VOLUME BY CYLINDRICAL SHELL METHOD 06_applications_of_the_integral-46.gif


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 y-axis 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 ≈
≈ π(x + Δx)2f(x) - πx2f(x) =
= π(x2 + 2x Δx + (Δx)2 - x2) f(x) =
= π(2x Δx + (Δx)2) f(x) ≈
≈ π2x Δx f(x),


ΔV ≈ 2πx f(x) Δx (compared to Δx).

By the Infinite Sum Theorem,


Example 3


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 y-axis. The volume of S can be found by subtracting the volume of the solid S1 generated by the region under y = f(x) from the volume of the solid S2 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 06_applications_of_the_integral-55.gif


Figure 6.2.12 (d)

Last Update: 2010-11-25