Solid of revolution

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis) that lies on the same plane.

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).

Rotating a curve

A representative disk is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length w) around some axis (located r units away), so that a cylindrical volume of π∫r²w units is enclosed.

Finding the volume

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.

Disc method

Disc integration about the y-axis.

Main article: Disk 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(x) and g(x) and the lines x = a and x = b about the x-axis is given by

$V = \pi \int_a^b \vert f^2(x) - g^2(x)\vert\,dx$

If g(x) = 0 (e.g. revolving an area between curve and x-axis), this reduces to:

$V = \pi \int_a^b f^2(x) \,dx \qquad (1)$

The method can be visualized by considering a thin vertical rectangle at x between y = f(x) on top and y = g(x) on the bottom, and revolving it about the x-axis; it forms a ring (or disc in the case that g(x) = 0), with outer radius f(x) and inner radius g(x). The area of a ring is π(R² − r²), where R is the outer radius (in this case f(x)), and r is the inner radius (in this case g(x)). Summing up all of the areas along the interval gives the total volume. Alternatively, where each disc has a radius of f(x), the discs approach perfect cylinders as their height dx approaches zero. The volume of each infinitesimal disc is therefore πf²(x)dx. An infinite sum of the discs between a and b manifests itself as integral (1).

Cylinder method

Shell integration.

Main article: Shell integration

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

$V = 2\pi \int_a^b x\vert f(x) - g(x)\vert\,dx$

If g(x) = 0 (e.g. revolving an area between curve and x-axis), this reduces to:

$V = 2\pi \int_a^b x \vert f(x) \vert \,dx$

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.

Parametric form

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, resp. the y-axis are given by

$V_{x} = \int_a^b \, \pi \, y^2 \, \frac{dx}{dt} \, dt$

$V_{y} = \int_a^b \pi \, \, x^2 \, \frac{dy}{dt} \, dt .$

Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the x-axis, resp. the y-axis are given by

$A_{x} = \int_a^b 2 \pi y \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

$A_{y} = \int_a^b 2 \pi x \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

See also

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

References

• CliffsNotes.com. Volumes of Solids of Revolution. 12 Apr 2011 <http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html>.
• Frank Ayres, Elliott Mendelson:Schaum's outlines: Calculus. McGraw-Hill Professional 2008, ISBN 9780071508612. pp. 244-248 (online copy at Google Books)
• Weisstein, Eric W., "Solid of Revolution" from MathWorld.

External links

• Plot a solid of revolution