We arrived at Formula 6.3.2, V = ∫b 2πx f(x) dx, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where f is one-to-one and therefore has an inverse function. Use the figure to show that




Make the substitution y = f(x) and then use integration by parts on the resulting integral to prove that V = ∫b 2πxf(x) dx.

V = mb'd ma'c - *r[9(y)]*dy %3D x= g(y) y= f(x) a.