Solid of revolution

Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness δx, or a cylindrical shell of width δx; and then find the limiting sum of these volumes as δx approaches 0, a value which may be found by evaluating a suitable integral. A more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration.

The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution.

The volume of the solid formed by rotating the area between the curves of f(y) and g(y) and the lines y = a and y = b about the y-axis is given by $${\displaystyle V=\pi \int _{a}^{b}\left|f(y)^{2}-g(y)^{2}\right|\,dy\,.}$$  If g(y) = 0 (e.g. revolving an area between the curve and the y-axis), this reduces to: $${\displaystyle V=\pi \int _{a}^{b}f(y)^{2}\,dy\,.}$$ 

The method can be visualized by considering a thin horizontal rectangle at y between f(y) on top and g(y) on the bottom, and revolving it about the y-axis; it forms a ring (or disc in the case that g(y) = 0), with outer radius f(y) and inner radius g(y). The area of a ring is π(R² − r²), where R is the outer radius (in this case f(y)), and r is the inner radius (in this case g(y)). The volume of each infinitesimal disc is therefore πf(y)² dy. The limit of the Riemann sum of the volumes of the discs between a and b becomes integral (1).

Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D): $${\displaystyle V=\iiint _{D}dV=\int _{a}^{b}\int _{g(z)}^{f(z)}\int _{0}^{2\pi }r\,d\theta \,dr\,dz=2\pi \int _{a}^{b}\int _{g(z)}^{f(z)}r\,dr\,dz=2\pi \int _{a}^{b}{\frac {1}{2}}r^{2}\Vert _{g(z)}^{f(z)}\,dz=\pi \int _{a}^{b}(f(z)^{2}-g(z)^{2})\,dz}$$ 

The shell method (sometimes referred to as the "cylinder method") is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the y-axis is given by $${\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx\,.}$$  If g(x) = 0 (e.g. revolving an area between curve and y-axis), this reduces to: $${\displaystyle V=2\pi \int _{a}^{b}x|f(x)|\,dx\,.}$$ 

The method can be visualized by considering a thin vertical rectangle at x with height f(x) − g(x), and revolving it about the y-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is 2πrh, where r is the radius (in this case x), and h is the height (in this case f(x) − g(x)). Summing up all of the surface areas along the interval gives the total volume.

This method may be derived with the same triple integral, this time with a different order of integration: $${\displaystyle V=\iiint _{D}dV=\int _{a}^{b}\int _{g(r)}^{f(r)}\int _{0}^{2\pi }r\,d\theta \,dz\,dr=2\pi \int _{a}^{b}\int _{g(r)}^{f(r)}r\,dz\,dr=2\pi \int _{a}^{b}r(f(r)-g(r))\,dr.}$$ 

When a curve is defined by its parametric form (x(t),y(t)) in some interval [a,b], the volumes of the solids generated by revolving the curve around the x-axis or the y-axis are given by^[1] $${\displaystyle {\begin{aligned}V_{x}&=\int _{a}^{b}\pi y^{2}\,{\frac {dx}{dt}}\,dt\,,\\V_{y}&=\int _{a}^{b}\pi x^{2}\,{\frac {dy}{dt}}\,dt\,.\end{aligned}}}$$ 

Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the x-axis or the y-axis are given by^[2] $${\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}2\pi y\,{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\,dt\,,\\A_{y}&=\int _{a}^{b}2\pi x\,{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\,dt\,.\end{aligned}}}$$ 

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

Computing the partial derivatives yields $${\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=\left\langle {\frac {dy}{dt}}\cos(\theta ),{\frac {dy}{dt}}\sin(\theta ),{\frac {dx}{dt}}\right\rangle ,}$$  $${\displaystyle {\frac {\partial \mathbf {r} }{\partial \theta }}=\left\langle -y\sin(\theta ),y\cos(\theta ),0\right\rangle }$$  and computing the cross product yields $${\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}=\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle =y\left\langle \cos(\theta ){\frac {dx}{dt}},\sin(\theta ){\frac {dx}{dt}},{\frac {dy}{dt}}\right\rangle }$$  where the trigonometric identity ${\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}$  was used. With this cross product, we get $${\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }\left\|y\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle \right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\cos ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\sin ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}2\pi y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ dt\end{aligned}}}$$  where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.

For a polar curve ${\displaystyle r=f(\theta )}$  where ${\displaystyle \alpha \leq \theta \leq \beta }$  and ${\displaystyle f(\theta )\geq 0}$ , the volumes of the solids generated by revolving the curve around the x-axis or y-axis are $${\displaystyle {\begin{aligned}V_{x}&=\int _{\alpha }^{\beta }\left(\pi r^{2}\sin ^{2}{\theta }\cos {\theta }\,{\frac {dr}{d\theta }}-\pi r^{3}\sin ^{3}{\theta }\right)d\theta \,,\\V_{y}&=\int _{\alpha }^{\beta }\left(\pi r^{2}\sin {\theta }\cos ^{2}{\theta }\,{\frac {dr}{d\theta }}+\pi r^{3}\cos ^{3}{\theta }\right)d\theta \,.\end{aligned}}}$$ 

The areas of the surfaces of the solids generated by revolving the curve around the x-axis or the y-axis are given $${\displaystyle {\begin{aligned}A_{x}&=\int _{\alpha }^{\beta }2\pi r\sin {\theta }\,{\sqrt {r^{2}+\left({\frac {dr}{d\theta }}\right)^{2}}}\,d\theta \,,\\A_{y}&=\int _{\alpha }^{\beta }2\pi r\cos {\theta }\,{\sqrt {r^{2}+\left({\frac {dr}{d\theta }}\right)^{2}}}\,d\theta \,,\end{aligned}}}$$ 

• Gabriel's Horn
• Guldinus theorem
• Pseudosphere
• Surface of revolution
• Ungula

• "Volumes of Solids of Revolution". CliffsNotes.com. 12 Apr 2011. Archived from the original on 2012-03-19.
• Ayres, Frank; Mendelson, Elliott (2008). Calculus. Schaum's Outlines. McGraw-Hill Professional. pp. 244–248. ISBN 978-0-07-150861-2. (online copy, p. 244, at Google Books)
• Weisstein, Eric W. "Solid of Revolution". MathWorld.