Disc integration

Part of a series of articles about

Fundamental theorem

Definitions
Derivative (generalizations) Differential infinitesimal of a function total
Concepts
Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem
Rules and identities
Sum Product Chain Power Quotient General Leibniz Faà di Bruno's formula

Integral

Definitions
Lists of integrals
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae

Series

Convergence tests
Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor
Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel

Vector

Theorems
Gradient Divergence Curl Laplacian Directional derivative Identities
Divergence Gradient Green's Kelvin–Stokes

Multivariable

Formalisms
Matrix Tensor Exterior Geometric
Definitions
Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix

Specialized

Disc integration, also known in integral calculus as the disc method, is a means of 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 the 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:

\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:

\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:

\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. 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.

R_{\mathrm {O} }(x)^{2}-R_{\mathrm {I} }(x)^{2}\not \equiv \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 each formula. If $h$ is the value of a horizontal axis, then the volume equals

\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 $y = -2 x + x 2$ and $y = x$ along the axis $y = 4$ , one would integrate as follows:

\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 further axis may not be that obvious. In the previous example, even though $y = x$ is further up than $y = -2 x + x 2$ , it is the inner axis since it is closer to $y = 4$

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.

References

"Volumes of Solids of Revolution". CliffsNotes.com. Retrieved July 8, 2014.
Weisstein, Eric Wolfgang. "Method of Disks". MathWorld. Retrieved July 12, 2013.

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, at Google Books. Retrieved July 12, 2013.)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Disc integration

Definition

Function of x

Function of y

Washer method

See also

References

Function of $x$

Function of $y$