Exercise 1790Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Spherical coordinates with a radial integrand

#spherical coordinates#radial#integration

Statement

Compute $\displaystyle I=\iiint_{B_R}e^{-\sqrt{x^2+y^2+z^2}}\,dx\,dy\,dz$ where $B_R$ is the ball of radius $R$ .

Hint

Spherical: integrand $e^{-r}$ , Jacobian $r^2\sin\varphi$ .

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Setup.
$I=\int_0^{2\pi}\int_0^\pi\int_0^R e^{-r}r^2\sin\varphi\,dr\,d\varphi\,d\theta=4\pi\int_0^R r^2e^{-r}\,dr.$

Compute. $\int_0^R r^2e^{-r}\,dr=2-(R^2+2R+2)e^{-R}$ (two integrations by parts).

Result. $I=4\pi[2-(R^2+2R+2)e^{-R}]$ .

Conclusion. $I=4\pi\left(2-(R^2+2R+2)e^{-R}\right)$ .