Use cylindrical coordinates to evaluate , where R is the solid enclosed by and .

Question

Use cylindrical coordinates to evaluate  , where R is the solid enclosed by  and  .

Quizplus · Accepted Answer

The region enclosed by the two given surfaces can be described as follows in cylindrical coordinates:
$$0\leq z\leq 4$$
$$0\leq 	heta\leq 2\pi$$
$$2\leq r\leq 4$$
Thus, we can evaluate the integral as follows:
$$\iiint_R z^3\ dV=\int_0^{2\pi}\int_2^4\int_0^4r\cdot z^3\ dz\ dr\ d	heta=\int_0^{2\pi}\int_2^4\frac{1}{4}r\cdot 4^4-\frac{1}{4}r\cdot 0^4\ dr\ d	heta$$
$$=\int_0^{2\pi}\int_2^4 64r\ dr\ d	heta=\int_0^{2\pi}\frac{64}{2}\cdot (4^2-2^2)\ d	heta=192\pi$$
Therefore, the correct answer is choice C.

Use Cylindrical Coordinates to Evaluate , Where R Is