91Ó°ÊÓ

Find the average value of the function over the given solid. The average value of a continuous function \(f(x, y, z)\) over a solid region \(Q\) is $$\frac{1}{V} \iint_{Q} \int f(x, y, z) d V$$ where \(V\) is the volume of the solid region \(Q\). \(f(x, y, z)=x+y\) over the solid bounded by the sphere \(x^{2}+y^{2}+z^{2}=2\)

Approximation (a) use a computer algebra system to approximate the iterated integral, and (b) use the program in Exercise 68 to approximate the iterated integral for the given values of \(m\) and \(n\). $$ \begin{aligned} &\int_{1}^{4} \int_{1}^{2} \sqrt{x^{3}+y^{3}} d x d y \\ &m=6, n=4 \end{aligned} $$

Describe regions that are vertically simple and regions that are horizontally simple.