Disk integration - Citizendia

Disk integration is a means of calculating the volume of a solid of revolution, when integrating along the axis of revolution. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by A calculation is a deliberate process for transforming one or more inputs into one or more results with variable change The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Mathematics, Engineering, and Manufacturing, a solid of revolution is a solid figure obtained by rotating a Plane curve around This method models the generated 3 dimensional shape as a "stack" of an infinite number of disks (of varying radius) of infinitesimal thickness. It is possible to use "washers" instead of "disks" (the washer method) to obtain "hollow" solids of revolutions, and uses the same principles that underlies disk integration.

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:

$\pi \int_a^b {\left[R(x)\right]}^2\ \mathrm{d}x$

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). A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

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:

$\pi \int_c^d {\left[R(y)\right]}^2\ \mathrm{d}y$

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). A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

"Hollow" solid of revolution

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

$\pi \int_a^b \left({\left[R_O(x)\right]}^2 - {\left[R_I(x)\right]}^2\right) \mathrm{d}x$

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. ${\left[R_O(x)\right]}^2 - {\left[R_I(x)\right]}^2\ \not\equiv \; {\left[R_O(x) - R_I(x)\right]}^2$

NOTE: the above 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 =

$\pi \int_a^b \left({\left[h-R_O(x)\right]}^2 - {\left[h-R_I(x)\right]}^2\right) \mathrm{d}x$

For example, to rotate the region between $y = - 2 x + x 2$ and $y = x$

along the axis $y = 4$ , you would have to integrate as follows:

$\pi \int_0^3 \left({\left[4-\left(-2x+x^2\right)\right]}^2 - {[4-x]}^2\right) \mathrm{d}x$

Note that when you integrate 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. You simply must solve each equation for $x$ before you plug them into the integration formula.

References

In Mathematics, Engineering, and Manufacturing, a solid of revolution is a solid figure obtained by rotating a Plane curve around Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating

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Definition

Function of x

Function of y

"Hollow" solid of revolution

See also

References