Chapter 5.1, Problem 54E

[T] Consider the function $f\left(x,y\right)=\sin \left(x^2\right)\cos \left(y^2\right)$ . where (x. y) $\left(x,y\right)\in R=\left[-1,1\right]\times \left[-1,1\right]$ .

a. Use the midpoint rule with m= n = 2. 4,. ... 10 to estimate the double integral $I={\displaystyle \underset{R}{\iint }\sin \left(x^2\right)\cos \left(y^2\right)d}A$ . Round your answers to the nearest hundredths.

b. For m = n = 2. find the average value of f over the region R. Round your answer to the nearest hundredths.

c. Use a CAS to graph in the same coordinate system the solid whose volume is given by ${\displaystyle \underset{R}{\iint }\sin \left(x^2\right)\cos \left(y^2\right)dA}$ and the plane $z=f_{ave}$