Difference between revisions of "Integration using polar coordinates"
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This can be integrated by separating the integral. The <math>\phi</math> integral gives <math>2 \pi</math> and the <math>r</math> integral gives <math>\text{1}</math>. This means the initial integral, <math>I</math> is <math>\sqrt{2 \pi}</math>. | This can be integrated by separating the integral. The <math>\phi</math> integral gives <math>2 \pi</math> and the <math>r</math> integral gives <math>\text{1}</math>. This means the initial integral, <math>I</math> is <math>\sqrt{2 \pi}</math>. | ||
+ | |||
+ | == Other methods of integration == | ||
+ | The primary methods of integration include: | ||
+ | |||
+ | * [[Integration by parts]] | ||
+ | * [[Integration using polar coordinates]] | ||
+ | * [[Multiple integration]] | ||
+ | * [[Residue calculus]] (for definite integrals) | ||
+ | * [[Method of simultaneous convolutions]] | ||
+ | * [[Mellin transform]]s | ||
+ | * [[Inflation-restriction sequence]]s | ||
+ | |||
+ | [[Category:Calculus]] |
Revision as of 14:54, 14 December 2016
Integration using polar coordinates is a technique for solving integrals using polar coordinates. Sometimes an integral that is complicated in one set or coordinates, such as Cartesian coordinates become very easy or even trivial in polar coordinates.
Example
Consider the integral
This can be converted into polar coordinated by multiplying by itself, so that
This can be expressed as the double integral,
Now the usefulness of polar coordinates becomes apparent as in polar coordinates, . The bounds mean the integral is over the entire x-y plane, so varies from to and from to . To convert the differentials, we must multiply by the Jacobian, to get
This can be integrated by separating the integral. The integral gives and the integral gives . This means the initial integral, is .
Other methods of integration
The primary methods of integration include:
- Integration by parts
- Integration using polar coordinates
- Multiple integration
- Residue calculus (for definite integrals)
- Method of simultaneous convolutions
- Mellin transforms
- Inflation-restriction sequences