Chapter 14.8, Problem 40ES

Use the Wallis sine and cosine formulas:

$\begin{array}{l}{\displaystyle {\int }_0^{\pi /2}{\sin }^{n}xdx}=\frac{\pi }{2}\cdot \frac{1\cdot 3\cdot 5\cdot \cdot \cdot \left(n-1\right)}{2\cdot 4\cdot 6\cdot \cdot \cdot n}\left(\begin{array}{l}n\text{even}\\ \text{and }\ge \text{2}\end{array}\right)\\ {\displaystyle {\int }_0^{\pi /2}{\sin }^{n}xdx}=\frac{2\cdot 4\cdot 6\cdot \cdot \cdot \left(n-1\right)}{3\cdot 5\cdot 7\cdot \cdot \cdot n}\left(\begin{array}{l}n\text{odd}\\ \text{and }\ge \text{3}\end{array}\right)\\ {\displaystyle {\int }_0^{\pi /2}{\cos }^{n}xdx}=\frac{\pi }{2}\cdot \frac{1\cdot 3\cdot 5\cdot \cdot \cdot \left(n-1\right)}{2\cdot 4\cdot 6\cdot \cdot \cdot n}\left(\begin{array}{l}n\text{even}\\ \text{and }\ge \text{2}\end{array}\right)\\ {\displaystyle {\int }_0^{\pi /2}{\cos }^{n}xdx}=\frac{2\cdot 4\cdot 6\cdot \cdot \cdot \left(n-1\right)}{3\cdot 5\cdot 7\cdot \cdot \cdot n}\left(\begin{array}{l}n\text{odd}\\ \text{and }\ge 3\end{array}\right)\end{array}$

Find the mass of the solid in the first octant bounded above by the paraboloid $z=4-x^2-y^2,$ below by the plane $z=0,$ and laterally by the cylinder $x^2+y^2=2x$ and the plane $y=0$ assuming the density to be $\delta \left(x,y,z\right)=z.$