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

$\text{As we have}\\ x^{2}+y^{2}+z^{2} \leq r^{2}\\ we\ get\\ \rho=0 \rightarrow r ,\\ \phi=0 \rightarrow \pi \text { and } \\ \theta=0 \rightarrow 2 \pi \\ \text { So, we obtain} \\ \iiint\limits_{D} c d V=\int_{0}^{2 \pi} \int_{0}^{\pi} \int_{0}^{r} c \rho^{2} \sin (\phi) d \rho d \phi d \theta \\ =-\left.\left.\left.c \frac{\rho^{3}}{3}\right|_{0} ^{r} \cos (\phi)\right|_{0} ^{\pi} \theta\right|_{0} ^{2 \pi}\\ =-c \frac{r^{3}}{3}(\cos (\pi)-\cos (0)(2 \pi-0))\\ =-c \frac{r^{3}}{3}(-2)(2 \pi) \\ =\frac{4}{3} \pi c r^{3}\\ \text{Then, the correct answer is (b)}$