Shell Method
The Shell method is a method of computing the volume of a surface of revolution. It is also commonly referred to as the method of cylindrical shells. This method is most convenient when, for instance, a region in the x,y-plane below the graph of a function of x is being revolved about the y-axis, or a region below the graph of a function of x is being revolved about the y-axis. In such cases, it is usually preferable to the most familiar method, the disk method.
To derive the shell-method formula, first suppose that we have a nonnegative continuous function f(x) defined on an interval [a,b]. (The derivation is analogous for a non-positive function, or for a function of y instead of x.) Suppose that we are revolving the area below the graph of f (and above the x-axis) around the y-axis, and wish to compute the volume of the resulting region.
As always when using the techniques of calculus to measure some geometric quantity (area, arc-length, volume, etc.), we begin with an approximation. Divide the interval [a,b] into subintervals: a = x0 < x1 < x2 < ... < xn-1 < xn = b. We restrict our attention to one subinterval [xk-1,xk], and try to approximate the volume that results from revolving around the y-axis only the region below the graph of f and between xk-1 and xk.
Suppose for the moment that the function f has the constant value f(xk) on this interval. Then the surface of revolution is in fact a cylindrical shell (that is, the volume between two concentric cylinders, one with radius xk-1 and one with radius xk). The height of this shell is f(xk), and so its volume is p(xk2 - xk-12)f(xk). If we follow the same procedure over each interval, we obtain the approximation
Sumk=1n (xk2 - xk-12)f(xk),
or, after factoring the difference of two squares and rearranging terms,
Sumk=1n f(xk) (xk + xk-1) (xk - xk-1)
This is, of course, not accurate for most functions, because in general the function f will not be constant over the interval [xk-1,xk]. However, if we allow the length of this interval to shrink, the approximation will become more and more accurate, because the function f will vary less over a small interval than over a larger interval. Therefore as we let the number of subintervals approach infinity, the sum above approaches the actual volume. But that sum is a Riemann sum for the integral
(Int)ab f(x)(2x)dx.
Thus this integral gives the volume of the surface of revolution.
It is plain to see that this formula is more manageable than the disk formula: first and most importantly, to use the disk method we would have to compute the inverse function of f (which can be difficult or, occasionally, even impossible). We must also consider the region being rotated as the area between two curves, thus requiring us to compute two integrals instead of one (one simple, but one possibly difficult).
As with the disk method, the shell method can be generalized to the case in which the region being rotated is the area between the graphs of two functions f and g. The generalized formula is (assuming the graph of f lies above the graph of g)
(Int)ab 2 x (f(x) - g(x))dx.
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