Question

Set up a double integral in rectangular coordinates for calculating the volume of the solid under the graph of the function f(x,y) = 32 - x^2 - y^2 and above the plane z = 7.

(include limits of integration)

Answer 1

Well we have:
\[7 \le z \le 32-x^2-y^2 \implies 7 \le z \le 32 - r^2\]
So from that we have:
\[7=32-r^2 \implies r^2=25 \implies r=5 \implies 0 \le r \le 5; 0 \le \phi \le 2 \pi\]
So we have:
\[V=\int\limits_0^{2 \pi} \int\limits_0^5 \int\limits_7^{32-r^2} r dz dr d \phi\]