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 , or a cylindrical shell of width ; and then find the limiting sum of these volumes as 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 method 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 and and the lines and about the -axis is given by V = \pi \int_a^b \left| f(y)^2 - g(y)^2\right|\,dy\, . If (e.g. revolving an area between the curve and the -axis), this reduces to: V = \pi \int_a^b f(y)^2 \,dy\, . The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius . The area of a ring is , where is the outer radius (in this case ), and is the inner radius (in this case ). The volume of each infinitesimal disc is therefore . The limit of the Riemann sum of the volumes of the discs between and 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): 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 Shell Method of 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 and and the lines and about the -axis is given by V = 2\pi \int_a^b x |f(x) - g(x)|\, dx\, . If (e.g. revolving an area between curve and -axis), this reduces to: V = 2\pi \int_a^b x | f(x) | \,dx\, . The method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it forms a cylindrical shell. The lateral surface area of a cylinder is , where is the radius (in this case ), and is the height (in this case ). 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: 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. ==Parametric form==

: study of a vase as a solid of revolution by Paolo Uccello. 15th century When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by{{cite book \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} Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given by{{cite book \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} This can also be derived from multivariable integration. If a plane curve is given by \langle x(t), y(t) \rangle then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by \mathbf{r}(t, \theta) = \langle y(t)\cos(\theta), y(t)\sin(\theta), x(t)\rangle with 0 \leq \theta \leq 2\pi. Then the surface area is given by the surface integral 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 \frac{\partial \mathbf{r}}{\partial t} = \left\langle \frac{dy}{dt} \cos(\theta), \frac{dy}{dt} \sin(\theta), \frac{dx}{dt} \right\rangle, \frac{\partial \mathbf{r}}{\partial \theta} = \left\langle -y \sin(\theta), y \cos(\theta), 0 \right\rangle and computing the cross product yields \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 \sin^2(\theta) + \cos^2(\theta) = 1 was used. With this cross product, we get \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\| \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} where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar. == Polar form ==

For a polar curve r=f(\theta) where \alpha\leq \theta\leq \beta and f(\theta) \geq 0, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are \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} The areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given \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} ==See also==