Question

Find the volume of sphere by triple integration.

(a) \(8a^3 \frac{π}{3} \)

(b) \(4a^3 \frac{π}{3} \)

(c) \(2a^3 \frac{π}{3} \)

(d) \(a^3 \frac{π}{3} \)

This question was posed to me in class test.

My doubt is from Volume Integrals in section Vector Integral Calculus of Engineering Mathematics

Answer 1

The correct answer is (b) \(4a^3 \frac{π}{3} \)

For explanation: The sphere is given by x^2+y^2+z^2=a^2.

θ : 0 to 2 π

φ : 0 to  π

r :  0 to a

Volume = \( \int_0^a \int_0^π \int_0^{2π} r^2 sinθdθdrdϕ \)

\(= 4(\int_0^a r^2 dr)(\int_0^π sinθdθ)(\int_0^{π/2}dϕ) \)

\(= 4a^3 \frac{π}{3}.\)