Disc integration, also known inintegral calculus as thedisc method, is a method for calculating thevolume of asolid of revolution of a solid-state material whenintegrating along an axisparallel to theaxis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs (the "washer method") to obtain hollow solids of revolution. This is in contrast toshell integration, which integrates along an axisperpendicular to the axis of revolution.

If the function to be revolved is a function ofx, the following integral represents the volume of the solid of revolution:

${\displaystyle \pi {\int }_a^{b}R(x{)}^{2}dx}$

whereR(x) is the distance between the function and the axis of rotation. This works only if theaxis of rotation is horizontal (example:y = 3 or some other constant).

If the function to be revolved is a function ofy, the following integral will obtain the volume of the solid of revolution:

${\displaystyle \pi {\int }_c^{d}R(y{)}^{2}dy}$

whereR(y) is the distance between the function and the axis of rotation. This works only if theaxis of rotation is vertical (example:x = 4 or some other constant).

To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. This can be calculated in a single integral similar to the following:

${\displaystyle \pi {\int }_a^b\left(R_{\mathrm{O}}(x{)}^2-R_{\mathrm{I}}(x{)}^2\right)dx}$

whereR_O(x) is the function that is furthest from the axis of rotation andR_I(x) is the function that is closest to the axis of rotation. For example, the next figure shows the rotation along thex-axis of the red "leaf" enclosed between the square-root and quadratic curves:

The volume of this solid is:

${\displaystyle \pi {\int }_0^1\left({\left(\sqrt{x}\right)}^2-{\left(x^2\right)}^2\right)dx.}$

One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.

${\displaystyle R_{\mathrm{O}}(x{)}^2-R_{\mathrm{I}}(x{)}^2\ne {\left(R_{\mathrm{O}}(x)-R_{\mathrm{I}}(x)\right)}^2}$

(This formula only works for revolutions about thex-axis.)

To rotate about any horizontal axis, simply subtract from that axis from each formula. Ifh is the value of a horizontal axis, then the volume equals

${\displaystyle \pi {\int }_a^b\left({\left(h-R_{\mathrm{O}}(x)\right)}^2-{\left(h-R_{\mathrm{I}}(x)\right)}^2\right)dx.}$

For example, to rotate the region betweeny = −2x +x² andy =x along the axisy = 4, one would integrate as follows:

${\displaystyle \pi {\int }_0^3\left({\left(4-\left(-2x+x^2\right)\right)}^2-(4-x{)}^2\right)dx.}$

The bounds of integration are the zeros of the first equation minus the second. Note that when integrating along an axis other than thex, the graph of the function which is furthest from the axis of rotation may not be obvious. In the previous example, even though the graph ofy =x is, with respect to the x-axis, further up than the graph ofy = −2x +x², with respect to the axis of rotation the functiony =x is the inner function: its graph is closer toy = 4 or the equation of the axis of rotation in the example.

The same idea can be applied to both they-axis and any other vertical axis. One simply must solve each equation forx before one inserts them into the integration formula.

• Solid of revolution
• Shell integration

• "Volumes of Solids of Revolution".CliffsNotes.com. RetrievedJuly 8, 2014.
• Weisstein, Eric W."Method of Disks".MathWorld.
• Frank Ayres,Elliott Mendelson.Schaum's Outlines: Calculus. McGraw-Hill Professional 2008,ISBN 978-0-07-150861-2. pp. 244–248 (online copy, p. 244, atGoogle Books. Retrieved July 12, 2013.)

• Integral calculus
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