Part of a series of articles about
Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellaneous Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis 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 revolutions. This is in contrast to shell integration, which integrates along an axis perpendicular to the axis of revolution.

Definition

Function of x

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

${\displaystyle \pi \int _{a}^{b}R(x)^{2}\,dx}$

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

Function of y

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

${\displaystyle \pi \int _{c}^{d}R(y)^{2}\,dy}$

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

Washer method

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:

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

where R_O(x) is the function that is farthest from the axis of rotation and R_I(x) is the function that is closest to the axis of rotation. For example, the next figure shows the rotation along the x-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.

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

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

To rotate about any horizontal axis, simply subtract from that axis from each formula. If h 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 between y = −2x + x² and y = x along the axis y = 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 the x, the graph of the function that is farthest from the axis of rotation may not be that obvious. In the previous example, even though the graph of y = x is, with respect to the x-axis, further up than the graph of y = −2x + x², with respect to the axis of rotation the function y = x is the inner function: its graph is closer to y = 4 or the equation of the axis of rotation in the example.

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