Question

Let c,r be constants, and D={(x,y,z):x^2+y^2+z^2≤r^2}. The answer to ∭D cdV

is
Select one:
a. (π^2cr^3)/3


b. (4πcr^3)/3

c. 4πcr^3

d. (πcr^4)/3


e. (4πcr^2)/2

f. (4r^3)/3


g. (πcr^3)/3

Answer 1

In spherical coordinates the volume element is $d\Omega=r^2\sin\theta d\theta d\phi$$dr$ then

$\iiint_{D}cdV=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\sin\theta d\theta\int_{0}^{r}cr^2dr=2\pi(-\cos\theta)|_0^\pi\frac{cr^3}{3}=$

$=\frac{4\pi cr^3}{3}$