Disk integration

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Topics in calculus

Fundamental theorem | Function | Limits of functions | Continuity | Mean value theorem | Vector calculus | Tensor calculus

Differentiation

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates

Integration

Integration by substitution | Integration by parts | Integration by trigonometric substitution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals

Contents 1 Introduction 2 Definition 2.1 Function of x 2.2 Function of y 2.3 "Hollow" Solid of Revolution 3 See also

Introduction

In calculus, the disc integration (or the disc method) is one of two popular methods (other is the "shell method") used in order to calculate the volume of a shape obtained by revolving the locus of a two dimensional equation around a straight axis (called the "axis of revolution"). Most commonly the axis of rotation is horizontal or vertical. This method models the generated 3 dimensional shape as a "stack" of an infinite number of cylinders (of varying radius) of infinitesimal thickness. One can think of it as stacking different sizes of coins on top of each other. This "stack" is called a "solid 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:

<math>\pi \int_{a}^{b} [R(x)]^2 dx</math>

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:

<math>\pi \int_{a}^{b} [R(y)]^2 dy</math>

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:

<math>\pi \int_{a}^{b} [R_O(x)]^2 - [R_I(x)]^2 dx</math>

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. <math>[R_O(x)]^2 - [R_I(x)]^2 \not\equiv \; [R_O(x) - R_I(x)]^2</math>

See also

• Shell Method