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

Correct option is (b).

Reason:

Given ,$c,r$ are constants.

Assume radius is $R$ instead of $r$ for the time being,then replace $R$ by $r$ in the final result.

$D$ is the region described as $D=\{(x,y,z):x^2+y^2+z^2≤r^2\}$ ,clearly $D$ contains all the points lies inside and surface of the sphere whose radius is $r$ .

Thus,

$$I=\int \int\int_{D}cdV=c\int \int\int_{D}dV$$

Let's draw the elemental volume

Hence,

$$I=c\int \int\int_{D}dV=c\int_{0}^{R}\int_{0}^{\pi}\int_{0}^{2\pi}r^2\sin(\theta)d\theta d\phi dr\\ \implies I=c\int_{0}^{R}\bigg(r^2\int_{0}^{\pi}\bigg(\sin(\theta)\int_{0}^{2\pi}d\phi \bigg)d\theta \bigg)dr=c\frac{4\pi R^3}{3}$$

Therefore the final answer is

$$I=c\frac{4\pi r^3}{3}$$