Chapter 5.5, Problem 252E

In the following exercises, the boundaries of the solid E are given in cylindrical coordinates.

a. Express the region E in cylindrical coordinates.

b. Convert the integral ${\displaystyle \underset{E}{\iiint }f\left(x,y,z\right)dV}$to cylindrical coordinates.

252. E is located in the first octant outside the circular

paraboloid z = 10 — 2r²and inside the cylinder $r=\sqrt{5}$and is bounded also by the plane z = 20 and $\theta =\frac{\pi }{4}$ .