Surface of revolution

Area formula

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 A_y is given by the integral $${\displaystyle A_y=2\pi {\int }_a^{b}x(t)\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt,}$$ provided that x(t) is never negative between the endpoints a and b. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity $${\displaystyle \sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt}$$ 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 $${\displaystyle A_x=2\pi {\int }_a^{b}y(t)\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt.}$$

If the continuous curve is described by the function y = f(x), a ≤ x ≤ b, then the integral becomes $${\displaystyle A_x=2\pi {\int }_a^{b}y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx=2\pi {\int }_a^{b}f(x)\sqrt{1+(f^{\prime }(x){)}^2}dx}$$ for revolution around the x-axis, and $${\displaystyle A_y=2\pi {\int }_a^{b}x\sqrt{1+{\left(\frac{dx}{dy}\right)}^2}dy}$$ for revolution around the y-axis (provided a ≥ 0). These come from the above formula.

This can also be derived from multivariable integration. If a plane curve is given by ${\displaystyle ⟨x(t),y(t)⟩}$ then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by ${\displaystyle \mathbf{r}(t,\theta )=⟨y(t)\cos (\theta ),y(t)\sin (\theta ),x(t)⟩}$ with ${\displaystyle 0\le \theta \le 2\pi }$. Then the surface area is given by the surface integral $${\displaystyle A_x={\iint }_{S}dS={\iint }_{[a,b]\times [0,2\pi ]}\Vert \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }t}\times \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }\Vert d\theta dt={\int }_a^b{\int }_0^{2\pi }\Vert \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }t}\times \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }\Vert d\theta dt.}$$

Computing the partial derivatives yields $${\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }t}=⟨\frac{dy}{dt}\cos (\theta ),\frac{dy}{dt}\sin (\theta ),\frac{dx}{dt}⟩,}$$ $${\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }=⟨-y\sin (\theta ),y\cos (\theta ),0⟩}$$ and computing the cross product yields $${\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }t}\times \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }=⟨y\cos (\theta )\frac{dx}{dt},y\sin (\theta )\frac{dx}{dt},y\frac{dy}{dt}⟩=y⟨\cos (\theta )\frac{dx}{dt},\sin (\theta )\frac{dx}{dt},\frac{dy}{dt}⟩}$$ where the trigonometric identity ${\displaystyle {\sin }^2(\theta )+{\cos }^2(\theta )=1}$ was used. With this cross product, we get $${\displaystyle \begin{array}{cc}{A}_x& ={\int }_a^b{\int }_0^{2\pi }\Vert \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }t}\times \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }\theta }\Vert d\theta dt\\ & ={\int }_a^b{\int }_0^{2\pi }\Vert y⟨y\cos (\theta )\frac{dx}{dt},y\sin (\theta )\frac{dx}{dt},y\frac{dy}{dt}⟩\Vert d\theta dt\\ & ={\int }_a^b{\int }_0^{2\pi }y\sqrt{{\cos }^2(\theta ){\left(\frac{dx}{dt}\right)}^2+{\sin }^2(\theta ){\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}d\theta dt\\ & ={\int }_a^b{\int }_0^{2\pi }y\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}d\theta dt\\ & ={\int }_a^{b}2\pi y\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt\end{array}}$$ 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 $${\displaystyle \begin{array}{cc}A& =2\pi {\int }_0^{\pi }\sin (t)\sqrt{(\cos (t){)}^2+(\sin (t){)}^2}dt\\ & =2\pi {\int }_0^{\pi }\sin (t)dt\\ & =4\pi .\end{array}}$$

For the case of the spherical curve with radius r, y(x) = √r² − x² rotated about the x-axis $${\displaystyle \begin{array}{cc}A& =2\pi {\int }_{-r}^r\sqrt{r^2-x^2}\sqrt{1+\frac{x^2}{r^2-x^2}}dx\\ & =2\pi r{\int }_{-r}^r\sqrt{r^2-x^2}\sqrt{\frac{1}{r^2-x^2}}dx\\ & =2\pi r{\int }_{-r}^{r}dx\\ & =4\pi r^2\end{array}}$$

A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area. A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.

There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.

Coordinate expressions

A surface of revolution given by rotating a curve described by ${\displaystyle y=f(x)}$ around the x-axis may be most simply described by ${\displaystyle y^2+z^2=f(x{)}^2}$. This yields the parametrization in terms of ${\displaystyle x}$ and ${\displaystyle \theta }$ as ${\displaystyle (x,f(x)\cos (\theta ),f(x)\sin (\theta ))}$. If instead we revolve the curve around the y-axis, then the curve is described by ${\displaystyle y=f(\sqrt{x^2+z^2})}$, yielding the expression ${\displaystyle (x\cos (\theta ),f(x),x\sin (\theta ))}$ in terms of the parameters ${\displaystyle x}$ and ${\displaystyle \theta }$.

If x and y are defined in terms of a parameter ${\displaystyle t}$, then we obtain a parametrization in terms of ${\displaystyle t}$ and ${\displaystyle \theta }$. If ${\displaystyle x}$ and ${\displaystyle y}$ are functions of ${\displaystyle t}$, then the surface of revolution obtained by revolving the curve around the x-axis is described by ${\displaystyle (x(t),y(t)\cos (\theta ),y(t)\sin (\theta ))}$, and the surface of revolution obtained by revolving the curve around the y-axis is described by ${\displaystyle (x(t)\cos (\theta ),y(t),x(t)\sin (\theta ))}$.

Geodesics

Meridians are always geodesics on a surface of revolution. Other geodesics are governed by Clairaut's relation.

Toroids

A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus.