Shell Method

The Method of Cylindrical Shells (Shell Method)
The shell method is a way of finding an exact value of the area of a solid of revolution. A solid of revolution is formed when a cross sectional strip(Figure 1) of a graph is rotated around the xy-plane. In the shell method, a strip that is parallel to the axis of rotation is taken so that when it is rotated about the xy-plane it forms acylindrical shell (Figure 2).

(Figure 1)

(Figure 2)

Figure 2 shows the graph of a parabola. The area bounded by the parabola and the x-axis is divided into cross sections. A cross section is thenrotated around the y-axis. When setting up problems, the cross section should be parallel to the axis of rotation. There are two general formulas for finding the volume by the shell method. One is usedwhen the axis of rotation is horizontal, and the other is used when the axis of rotation is vertical. The formulas are shown below. Axis of rotation is vertical V= Axis of rotation is horizontal V=2π ∫ x ⋅ f (x)dx
b a

2π ∫ b y ⋅ g( y)dy a

V is the volume, a and b are the limits of integration/bounds on the graph such that 0 ≤ a ≤ b , x/y is the distance of the cross section to the axis ofrotation, f(x)/g(y) is your function/equation. The main difference between the two is which variable to integrate with respect to. Example 1 The equation f ( x) = 3 − x revolved about the y-axis. 0 ≤x ≤ 3 .

Note: Sketching a graph of the function helps illustrate.

The problem asks to revolve about the y-axis, the strip is parallel to the y-axis. The limits of integration are given by theinformation; 0 ≤ x ≤ 3 . So a = 0 and b = 3. The distance from the strip to the y-axis is some unknown distance, x. f(x) is the given equation. V=
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2π ∫ 3 x ⋅ (3 − x)dx 0
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1

Example 2

The equation y = 3 − x revolved about the x-axis. 0 ≤ x ≤ 3 .

The strip is parallel to the axis of revolution. Writing the equation...