Calculating Volume Using Triple Integral

Question

I am trying to solve this problem but I am having difficulties to finish it. I would appreciate of someone can advice me on how to continue

Problem: Calculate $$\iiint_{V} Z\mathrm dV$$ where V is defined by $$ x^2+y^2 \le z^2 $$and$$ x^2+y^2+z^2 \le R^2 with R\gt0$$

Solution Using Cylindrical Coordinates $$\iiint_{V} Z\mathrm dV = \iiint_{V} Z\mathrm rdrdzd\theta $$ $$\iiint_{V} Z\mathrm dV = \iiint_{V} Z\mathrm rdrdzd\theta $$ $$x = rcos\theta, y= rsin\theta $$

Answer 1

HINT

• make some sketch for the region $E$ in $x-y$,$z-x/y$ plane to uderstand the shape of E
• define the low $r=f(z)$ and then the set up for the integral in the form

$$\int_0^{2\pi} d\theta \int_0^R dz \int_0^{f(z)} zrdr$$