Chapter 13.1, Problem 61E

A generalization Let R be as in Exercise 60, let F be an antiderivative of f with F(0) = 0, and let G be an antiderivative of F. Show that if f and F are integrable and r ≥ 1 and s ≥ 1 are real numbers, then

${\displaystyle \underset{R}{\iint }{x}^{2r-1}}{y}^{r-1}f\left(x^r{y}^5\right)dA=\frac{G\left(1\right)-G\left(0\right)}{rs}.$

60.    Two integrals Let $R=\left\{\left(x,y\right):0\le x\le 1,0\le y\le 1\right\}.$

a.    Evaluate ${\displaystyle \underset{R}{\iint }\cos \left(x\sqrt{y}\right)}dA.$

b.    Evaluate ${\displaystyle \underset{R}{\iint }{x}^{3}y\cos \left(x^2{y}^2\right)}dA.$