Let R be the region under the graph of y = (x) over the interval [a, a],

Let R be the region under the graph of y = ƒ(x) over the interval [−a, a], where ƒ(x) ≥ 0. Assume that R is symmetric with respect to the y-axis.

(a) Explain why y = ƒ(x) is even—that is, why ƒ(x) = ƒ(−x).

(b) Show that y = x ƒ(x) is an odd function.

(c) Use (b) to prove that M_y = 0.

(d) Prove that the COM of R lies on the y-axis (a similar argument applies to symmetry with respect to the x-axis).