Exercise 1791Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Volume of a sphere by triple integral

#spherical coordinates#volume#sphere

Statement

Compute the volume of the ball $B_R=\{x^2+y^2+z^2\le R^2\}$ using spherical coordinates.

Hint

$x=r\sin\varphi\cos\theta$ , $y=r\sin\varphi\sin\theta$ , $z=r\cos\varphi$ ; Jacobian $r^2\sin\varphi$ .

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Spherical. $0\le r\le R$ , $0\le\varphi\le\pi$ , $0\le\theta\le2\pi$ .

Jacobian. $r^2\sin\varphi$ .

Compute.
$V=\int_0^{2\pi}\int_0^\pi\int_0^R r^2\sin\varphi\,dr\,d\varphi\,d\theta=2\pi\cdot2\cdot\frac{R^3}{3}=\frac{4\pi R^3}{3}.$

Conclusion. $V=\frac{4\pi R^3}{3}$ .