stylistic:style_transfer, structural:reformatting
Volume

If the base of a circular cylinder has a radius r and the cylinder has height h, then its volume is given by V = π r 2 h {\displaystyle V=\pi r^{2}h}

This formula holds whether or not the cylinder is a right cylinder.

This formula may be established by using Cavalieri's principle.

In more generality, by the same principle, the volume of any cylinder is the product of the area of a base and the height. For example, an elliptic cylinder with a base having semi-major axis a, semi-minor axis b and height h has a volume V = Ah, where A is the area of the base ellipse (= π ab). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive x -axis and A (x) = A the area of each elliptic cross-section, thus: V = ∫ 0 h A (x) d x = ∫ 0 h π a b d x = π a b ∫ 0 h d x = π a b h. {\displaystyle V=\int _{0}^{h}A(x)dx=\int _{0}^{h}\pi abdx=\pi ab\int _{0}^{h}dx=\pi abh.}

Using cylindrical coordinates, the volume of a right circular cylinder can be calculated by integration V = ∫ 0 h ∫ 0 2 π ∫ 0 r s d s d ϕ d z = π r 2 h. {\displaystyle {\begin{aligned}V&=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{r}s\,\,ds\,d\phi \,dz\\&=\pi \,r^{2}\,h.\end{aligned}}}