In this exercise we will obtain a formula for the volume of the spherical wedge.

$($a$)$ Use a triple integral in cylindrical coordinates to show that the volume of the solid bounded above by a sphere $=$ $\_0,$ below by a cone $=$ $\_0,$ and on the sides by $=$ $\_1$ and $=$ $\_2($$\_1<$ $\_2)$ is V $=(1/3)($$\_0)^3(1$ cos $\_0)($$\_2$ $\_1)[$Hint: In cylindrical coordinates, the sphere has the equation r$^2+$ z$^2=($$\_0)^2$ and the cone has the equation z $=$ r cot $\_0.$ For simplicity, consider only the case $0<$ $\_0<$ $/2.$

$($b$)$ Subtract appropriate volumes and use the result in part $($a$)$ to deduce that the volume $$V of the spherical wedge is $$V$=((($p$\_2)^3-($p$\_1)^3)/3)($cos $\_1$ cos $\_2)$

$($c$)$ Apply the Mean$-$Value Theorem to the functions cos and $^3$ to deduce that the formula in part $($b$)$ can be written as $$V $=$ $^(2)$ sin $^$ $$$$ where $$ $=$ $\_2$ $\_1,$ $=$ $\_2$ $\_1,$ $=$ $\_2$ $\_1,$ and $^$ are between $\_1$ and $\_2,$ $^$ is between $\_1$ and $\_2.$