Calculus
Textbooks
Boundless Calculus
Derivatives and Integrals
Applications of Integration
Calculus Textbooks Boundless Calculus Derivatives and Integrals Applications of Integration
Calculus Textbooks Boundless Calculus Derivatives and Integrals
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 9
Created by Boundless

Volumes of Revolution

Disc and shell methods of integration can be used to find the volume of a solid produced by revolution.

Learning Objective

  • Distinguish between the disc and shell methods of integration in order to find the volumes of solids produced by revolution


Key Points

    • 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.
    • 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 shell 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.

Terms

  • cylinder

    a surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve

  • integration

    the operation of finding the region in the $xy$-plane bound by the function

  • revolution

    the turning of an object about an axis


Full Text

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 . Here, we will study how to compute volumes of these objects. 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 $\delta x$, or a cylindrical shell of width $\delta x$; and then find the limiting sum of these volumes as $\delta x$ approaches $0$, a value which may be found by evaluating a suitable integral.

A Volume of Revolution

A solid formed by rotating a curve around an axis.

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 $f(x)$ and $g(x)$ and the lines $x=a$ and $x=b$ about the $x$-axis is given by: 

$\displaystyle{V = \pi \int_a^b \left | f^2(x) - g^2(x) \right | \,dx}$

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

$\displaystyle{V = \pi \int_a^b f(x)^2 \,dx}$

Disc Integration

Disc integration about the $y$-axis. Integration is along the axis of revolution ($y$-axis in this case).

The method can be visualized by considering a thin horizontal rectangle at $y$between $y=f(x)$ on top and $y=g(x)$ on the bottom, and revolving it about the $y$-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:

 $\pi (R^2 - r^2)$

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:

 $\pi f^2(x) dx$

An infinite sum of the discs between $a$ and $b$ manifests itself as the integral seen above, replicated here:

$\displaystyle{V = \pi \int_a^b f(x)^2 \,dx}$

Shell Method

The shell 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:

$\displaystyle{V = 2\pi \int_a^b x \left | f(x) - g(x) \right | \,dx}$

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

 $\displaystyle{V = 2\pi \int_a^b x \left | f(x) \right | \,dx}$

Shell Integration

The integration (along the $x$-axis) is perpendicular to the axis of revolution ($y$-axis).

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 $2 \pi r h$, 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.

[ edit ]
Edit this content
Prev Concept
Work
Inverse Functions
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.