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

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),

whence 

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

Figure 6.2.12 (d)