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

$Since \ we \ have \\ x^2+y^2+z^2\leq r^2, \\ we \ obtain\ \\ \rho=0\rightarrow r ,\\ \phi=0\rightarrow \pi\ and \\ \theta=0\rightarrow 2\pi. \\ So, we \, get\\ \iiint\limits_D c\ dV\ =\int\limits_0^{2\pi}\int\limits_0^{\pi}\int\limits_0^{r} c\ \rho^2\ sin(\phi ) \ d\rho \ d\phi\ d\theta\\ \\ \\ \qquad \qquad\ = -c\ \frac{\rho^3}{3}|_0^{r}\ cos(\phi )\large|_0^{\pi}\theta|_0^{2\pi}\\ \qquad \quad\ \ =-c\ \frac{r^3}{3}(\ cos(\pi )-cos(0)(2\pi-0)\\ \qquad \quad\ \ =-c\ \frac{r^3}{3}(-2)(2\pi)\\ \qquad \quad\ \ = \frac{4}{3} \pi c r^3\\ \text{So, the correct answer is} \ \ \ b$