Disk integration

Disk integration is a means of calculating the volume of a solid of revolution, when integrating along the axis of revolution. This method models the generated 3 dimensional shape as a "stack" of an infinite number of disks (of varying radius) of infinitesimal thickness.

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:

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:

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).

"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:

Where RO(x) is the function that is farthest from the axis of rotation and RI(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.

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 is the value of a horizontal axis, then the volume =

For example, to rotate the region between and

along the axis , you would have to integrate as follows:

Note that when you integrate along an axis other than the , the further axis may not be that obvious. In the previous example, even though is further up than , it is the inner axis since it is closer to

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

References

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite
In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit L at a point p if the value of f(x) can be made as close to L as desired, by making x close enough to
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions[1]).
The term tensor has slightly different meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function.
secant joining the endpoints of the interval [ab] is parallel to the tangent at c.]] In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative
derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
In calculus, the product rule also called Leibniz's law (see derivation), governs the differentiation of products of differentiable functions.

It may be stated thus:

or in the Leibniz notation thus:

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 derivatives exist.

If the function one wishes to differentiate, , can be written as

In calculus, the chain rule is a formula for the derivative of the composite of two functions.

In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of
In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable.
Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point.
In differential calculus, related rates problems involve finding the rate at which a quantity is changing by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time.
The primary operation in differential calculus is finding a derivative. This table lists derivatives of many functions. In the following, f and g are differentiable functions, with respect to x, from the real numbers, and c is a real number.
INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
See the following pages for lists of integrals:
• List of integrals of rational functions
• List of integrals of irrational functions
• List of integrals of trigonometric functions
• List of integrals of inverse trigonometric 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 real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. The rule arises from the product rule of differentiation.
Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.

It makes use of the so-called "representative cylinder".
In calculus, the substitution rule is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, the substitution rule is a relatively important tool for mathematicians.
In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing the radical expressions:

A calculation is a deliberate process for transforming one or more inputs into one or more results.

The term is used in a variety of senses, from the very definite arithmetical calculation using an algorithm to the vague heuristics of calculating a strategy in a competition
The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.