Question

Using triple integral the volume of the sphere whose radius is ‘a’ unit is

Answer 1

Answer:

$\frac{4}{3} \pi a^3$

Step-by-step explanation:

Volume is given by triple intregral

$V=\int\int\int dV$.

Consider a small volume element (cubiodal) of length $da, ad\phi$ and$asin\phi d\theta.$

where  $\phi$ is azimuthal angle.

so,

$dV = (da)(ad\phi)(asin\phi d\theta)$.

where limit is

$0\leq a\leq a;\\0\leq \phi \leq \pi;\\0\leq \theta \leq 2\pi.$

Finally Volume equals to

$V = \int_0^{2\pi}\int_0^\pi sin\phi\int_0^a \,a^2da \,d\phi\, d\theta$,

$V = \frac{a^3}{3}\int_{\theta=0}^{\theta=2\pi} (-cos\phi)\bigg]_{0}^{\pi} d\theta =\frac{4}{3} \pi a^3$