Solid of Revolution

In the realm of geometry , a solid of revolution represents a three-dimensional solid figure generated by the rotation of a two-dimensional plane figure around a fixed straight line , known as the axis of revolution . This axis may or may not intersect the generatrix , except potentially at its boundary. The resulting surface that encloses the solid is referred to as the surface of revolution .

Fundamental Concepts

Volume Calculation via Pappus’s Centroid Theorem

When the rotating curve does not intersect the axis of revolution, the volume of the resulting solid can be determined using Pappus’s second centroid theorem . This theorem states that the volume is equivalent to the product of the area of the plane figure and the circumference of the circle described by its centroid during the rotation.

Representative Disc

A representative disc serves as a three-dimensional volume element within a solid of revolution. This element is formed by rotating a line segment of length w around an axis located at a distance r from the segment. The resulting cylindrical volume is given by the formula πr²w.

Methods for Volume Calculation

Disc Method

The disc method is employed when the slice of the solid is perpendicular to the axis of revolution, effectively integrating parallel to this axis.

Volume Formula

For a solid formed by rotating the area between the curves f(y) and g(y) and the lines y = a and y = b around the y-axis, the volume V is given by:

[ V = \pi \int_{a}^{b} \left| f(y)^2 - g(y)^2 \right| , dy ]

If g(y) = 0 (indicating rotation of an area between the curve and the y-axis), the formula simplifies to:

[ V = \pi \int_{a}^{b} f(y)^2 , dy ]

Visualization and Derivation

The disc method can be visualized by considering a thin horizontal rectangle at y between f(y) on top and g(y) on the bottom, rotating it around the y-axis to form a ring (or disc if g(y) = 0). The area of this ring is π(R² - r²), where R is the outer radius (f(y)) and r is the inner radius (g(y)). The volume of each infinitesimal disc is thus πf(y)²dy.

The derivation of the disc method can be justified using Fubini’s theorem and the multivariate change of variables formula. The volume V of the solid D can be expressed as:

[ 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} \left[ \frac{1}{2} r^2 \right]{g(z)}^{f(z)} , dz = \pi \int{a}^{b} (f(z)^2 - g(z)^2) , dz ]

Shell Method of Integration

The shell method , also known as the “cylinder method,” is utilized when the slice of the solid is parallel to the axis of revolution, integrating perpendicular to this axis.

Volume Formula

For a solid formed by rotating the area between the curves f(x) and g(x) and the lines x = a and x = b around the y-axis, the volume V is given by:

[ V = 2\pi \int_{a}^{b} x \left| f(x) - g(x) \right| , dx ]

If g(x) = 0 (indicating rotation of an area between the curve and the x-axis), the formula simplifies to:

[ V = 2\pi \int_{a}^{b} x \left| f(x) \right| , dx ]

Visualization and Derivation

The shell method can be visualized by considering a thin vertical rectangle at x with height f(x) - g(x), rotating it around the y-axis to form a cylindrical shell. The lateral surface area of this cylinder is 2πrh, where r is the radius (x) and h is the height (f(x) - g(x)).

The derivation of the shell method can also be justified using a triple integral 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

When a curve is defined by its parametric equations (x(t), y(t)) over an interval [a, b], the volumes of the solids generated by revolving the curve around the x-axis or the y-axis are given by:

[ V_x = \int_{a}^{b} \pi y^2 \frac{dx}{dt} , dt ] [ V_y = \int_{a}^{b} \pi x^2 \frac{dy}{dt} , dt ]

The areas of the surfaces of the solids generated by revolving the curve around the x-axis or the y-axis are given by:

[ 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 ]

Derivation from Multivariable Integration

For a plane curve given by (x(t), y(t)), the 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 ). 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 ]

The cross product is:

[ \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 ]

Using the trigonometric identity ( \sin^2(\theta) + \cos^2(\theta) = 1 ), we get:

[ 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 = \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 ]

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:

[ 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 ]

The areas of the surfaces of the solids generated by revolving the curve around the x-axis or the y-axis are given by:

[ 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 ]

Historical and Artistic Context

The study of solids of revolution has been a significant aspect of mathematics and art . For instance, the 15th-century artist Paolo Uccello explored the concept of a vase as a solid of revolution, demonstrating the intersection of mathematical principles and artistic expression.

Related Concepts

• Cylindrical symmetry • Gabriel’s Horn • Guldinus theorem • Pseudosphere • Surface of revolution • Ungula

Notes

• Sharma, A. K. (2005). Application Of Integral Calculus. Discovery Publishing House. p. 168. ISBN 81-7141-967-4. • Singh, Ravish R. (1993). Engineering Mathematics (6th ed.). Tata McGraw-Hill. p. 6.90. ISBN 0-07-014615-2.