Exercise 1787Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Volume using cylindrical coordinates

#cylindrical coordinates#volume#cone

Statement

Compute the volume of the solid bounded by $z=0$ , $z=h$ and $x^2+y^2\le R^2$ .

Hint

Cylindrical: $x=r\cos\theta$ , $y=r\sin\theta$ , $z=z$ ; Jacobian $r$ .

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Cylindrical. $0\le r\le R$ , $0\le\theta\le2\pi$ , $0\le z\le h$ .

Volume. $V=\int_0^{2\pi}\int_0^R\int_0^h r\,dz\,dr\,d\theta=2\pi\cdot\frac{R^2}{2}\cdot h=\pi R^2h$ .

Conclusion. $V=\pi R^2h$ .