Solid of revolution

Rotating a curve. The surface formed is a surface of revolution; it encloses a solid of revolution.

Solids of revolution (Matemateca Ime-Usp)

In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The surface created by this revolution and which bounds the solid is the surface of revolution.

Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem).

A representative disc is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length Template:Mvar) around some axis (located Template:Mvar units away), so that a cylindrical volume of Template:Math units is enclosed.

Finding the volumeEdit

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 Template:Mvar, or a cylindrical shell of width Template:Mvar; and then find the limiting sum of these volumes as Template:Mvar 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.

Disc methodEdit

File:Disc integration.svg

Disc integration about the y-axis

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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 Template:Math and Template:Math and the lines Template:Math and Template:Math about the Template:Mvar-axis is given by <math display="block">V = \pi \int_a^b \left| f(y)^2 - g(y)^2\right|\,dy\, .</math> If Template:Math (e.g. revolving an area between the curve and the Template:Mvar-axis), this reduces to: <math display="block">V = \pi \int_a^b f(y)^2 \,dy\, .</math>

The method can be visualized by considering a thin horizontal rectangle at Template:Mvar between Template:Math on top and Template:Math on the bottom, and revolving it about the Template:Mvar-axis; it forms a ring (or disc in the case that Template:Math), with outer radius Template:Math and inner radius Template:Math. The area of a ring is Template:Math, where Template:Mvar is the outer radius (in this case Template:Math), and Template:Mvar is the inner radius (in this case Template:Math). The volume of each infinitesimal disc is therefore Template:Math. The limit of the Riemann sum of the volumes of the discs between Template:Mvar and Template:Mvar 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): <math display="block">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^{f(z)}_{g(z)} \,dz = \pi \int_a^b (f(z)^2 - g(z)^2)\,dz</math>

Shell Method of IntegrationEdit

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File:Shell integration.svg

Shell integration

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 Template:Math and Template:Math and the lines Template:Math and Template:Math about the Template:Mvar-axis is given by <math display="block">V = 2\pi \int_a^b x |f(x) - g(x)|\, dx\, .</math> If Template:Math (e.g. revolving an area between curve and Template:Mvar-axis), this reduces to: <math display="block">V = 2\pi \int_a^b x | f(x) | \,dx\, .</math>

The method can be visualized by considering a thin vertical rectangle at Template:Mvar with height Template:Math, and revolving it about the Template:Mvar-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is Template:Math, where Template:Mvar is the radius (in this case Template:Mvar), and Template:Mvar is the height (in this case Template:Math). 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: <math display="block">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.</math>

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Parametric formEdit

File:Paolo uccello, studio di vaso in prospettiva 02.jpg

Mathematics and art: study of a vase as a solid of revolution by Paolo Uccello. 15th century

When a curve is defined by its parametric form Template:Math in some interval Template:Math, the volumes of the solids generated by revolving the curve around the Template:Mvar-axis or the Template:Mvar-axis are given by<ref>Template:Cite book</ref> <math display="block">\begin{align} 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{align}</math>

Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the Template:Mvar-axis or the Template:Mvar-axis are given by<ref>Template:Cite book</ref> <math display="block">\begin{align} 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{align}</math>

This can also be derived from multivariable integration. If a plane curve is given by <math>\langle x(t), y(t) \rangle</math> then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by <math>\mathbf{r}(t, \theta) = \langle y(t)\cos(\theta), y(t)\sin(\theta), x(t)\rangle</math> with <math>0 \leq \theta \leq 2\pi</math>. Then the surface area is given by the surface integral <math display="block">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.</math>

Computing the partial derivatives yields <math display="block">\frac{\partial \mathbf{r}}{\partial t} = \left\langle \frac{dy}{dt} \cos(\theta), \frac{dy}{dt} \sin(\theta), \frac{dx}{dt} \right\rangle,</math> <math display="block">\frac{\partial \mathbf{r}}{\partial \theta} = \left\langle -y \sin(\theta), y \cos(\theta), 0 \right\rangle</math> and computing the cross product yields <math display="block">\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 </math> where the trigonometric identity <math>\sin^2(\theta) + \cos^2(\theta) = 1</math> was used. With this cross product, we get <math display="block">\begin{align} 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{align}</math> where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.

Polar formEdit

For a polar curve <math>r=f(\theta)</math> where <math>\alpha\leq \theta\leq \beta</math> and <math>f(\theta) \geq 0</math>, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are <math display="block">\begin{align} 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{align}</math>

The areas of the surfaces of the solids generated by revolving the curve around the Template:Mvar-axis or the Template:Mvar-axis are given <math display="block">\begin{align} 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{align}</math>