Chapter 9.1, Problem 7CP

(a) Use calculus to evaluate the integral ${\displaystyle {\int }_0^1{\displaystyle {\int }_{x^2}^{\sqrt{x}}}}$, ${\displaystyle {\int }_0^1{\displaystyle {\int }_{x^2}^{\sqrt{x}}xydydx}}$ . (b) Use $n={10}^6$ pairs in the unit square $\left[0,1\right]\times \left[0,1\right]$

to estimate the integral as a Type 1 Monte Cub problem. (Average the function that is equal to xy if $(x,y)$ is in the integration domain and 0 if not)