(a) Let f(x) be an even function of x so that f(x) = f (-x). Show that ƒa-a ƒ (x) dx = 2 ƒ a0 ƒ (x) dx. (Hint: Write the integral from -a to a as the sum of the integral from -a to 0 and the integral from 0 to a. In the first integral, make the change of variable x' = -x.)
(b) Let g(x) be an odd function of x so that g(x) = -g (-x). Use the method given in the hint for part (a) to show that ƒ a-a g (x) dx = 0.
(c) Use the result of part (b) to show why Ey in Example 21.11 (Section 21.5) is zero.