4.

(a) Suppose that E is a two-dimensional region and S = {(x, y, z) E R3 : (x, y) E E and z = o}. Prove that Area (E) = lis dO"

Let I: [a, b] -+ R be a CP function, let C be the curve in R2 determined by z = I(x), a :::; x :::;

b, and let 8 be the surface in R3 determined by z = I(x), a:::; x :::;

b, c :::; y :::;

d. Show that a(8) = (d - c)L(C).

(c) Let 1 : [a, b] -+ R be a CP function and let 8 be the surface obtained by revolving the curve y = I(x), a :::; x :::;

b, about the x axis. Prove that the surface area of 8 is a(8) = 27r lb I/(x)lvl + 1f'(x)12 dx.