Editing Cylinder (section)

===Volume===
If the base of a circular cylinder has a [[radius]]  {{math|''r''}} and the cylinder has height {{mvar|h}}, then its [[volume]] is given by
<math display=block>V = \pi r^2h</math>

This formula holds whether or not the cylinder is a right cylinder.{{sfn|Wentworth|Smith|1913|p=359}}

This formula may be established by using [[Cavalieri's principle]].
[[File:Elliptic cylinder abh.svg|thumb|A solid elliptic right cylinder with the semi-axes {{math|''a''}} and {{math|''b''}} for the base ellipse and height {{math|''h''}}]]
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 and semi-minor axes|semi-major axis]] {{mvar|a}}, semi-minor axis {{mvar|b}} and height {{mvar|h}} has a volume {{math|1=''V'' = ''Ah''}}, where {{mvar|A}} is the area of the base ellipse (= {{math|{{pi}}''ab''}}). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive {{mvar|x}}-axis and {{math|1=''A''(''x'') = ''A''}} the area of each elliptic cross-section, thus:
<math display=block>V = \int_0^h A(x) dx = \int_0^h \pi ab dx = \pi ab \int_0^h dx = \pi a b h.</math>

Using [[cylindrical coordinates]], the volume of a right circular cylinder can be calculated by integration
<math display=block>\begin{align}
V &= \int_0^h \int_0^{2\pi} \int_0^r s \,\, ds \, d\phi \, dz \\[5mu]
&= \pi\,r^2\,h.
\end{align}</math>