If the curve is described by the parametric functions x(t), y(t), with t ranging over some interval [a,b], and the axis of revolution is the y-axis, then the surface area Ay is given by the integral
provided that x(t) is never negative between the endpoints a and b. This formula is the calculus equivalent of Pappus's centroid theorem.[3] The quantity
comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity 2πx(t) is the path of (the centroid of) this small segment, as required by Pappus' theorem.
Likewise, when the axis of rotation is the x-axis and provided that y(t) is never negative, the area is given by[4]
If the continuous curve is described by the function y = f(x), a ≤ x ≤ b, then the integral becomes
for revolution around the x-axis, and
for revolution around the y-axis (provided a ≥ 0). These come from the above formula.[5]
This can also be derived from multivariable integration. If a plane curve is given by 
then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by 
with 
. Then the surface area is given by the surface integral
![{\displaystyle A_{x}=\iint _{S}dS=\iint _{[a,b]\times [0,2\pi ]}\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt.}](//wikimedia.org/api/rest_v1/media/math/render/svg/9b6e20c212de99b253a52f3442b8f74ea73d61ac)
Computing the partial derivatives yields
and computing the cross product yields
where the trigonometric identity 
was used. With this cross product, we get
![{\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }\left\|y\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle \right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\cos ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\sin ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}2\pi y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ dt\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/67ad143220b39defd5b72caa82d8beb7b47d4946)
where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.
For example, the spherical surface with unit radius is generated by the curve y(t) = sin(t), x(t) = cos(t), when t ranges over [0,π]. Its area is therefore
For the case of the spherical curve with radius r, y(x) = √r2 − x2 rotated about the x-axis
A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area.[6] A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.[6]
There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[7]
A surface of revolution given by rotating a curve described by 
around the x-axis may be most simply described by 
. This yields the parametrization in terms of 
and 
as 
. If instead we revolve the curve around the y-axis, then the curve is described by 
, yielding the expression 
in terms of the parameters and .
If x and y are defined in terms of a parameter 
, then we obtain a parametrization in terms of and . If and 
are functions of , then the surface of revolution obtained by revolving the curve around the x-axis is described by 
, and the surface of revolution obtained by revolving the curve around the y-axis is described by 
.
![{\displaystyle A_{x}=\iint _{S}dS=\iint _{[a,b]\times [0,2\pi ]}\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/9b6e20c212de99b253a52f3442b8f74ea73d61ac)
![{\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }\left\|y\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle \right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\cos ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\sin ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}2\pi y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ dt\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/67ad143220b39defd5b72caa82d8beb7b47d4946)
