Calculus/Polar Integration

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Contents

IntroductionEdit

Integrating a polar equation requires a different approach than integration under the Cartesian system, hence yielding a different formula, which is not as straightforward as integrating the function   .

ProofEdit

In creating the concept of integration, we used Riemann sums of rectangles to approximate the area under the curve. However, with polar graphs, one can use sectors of circles with radius r and angle measure dθ. The area of each sector is then (πr²)(dθ/2π) and the sum of all the infinitesimally small sectors' areas is:   , This is the form to use to integrate a polar expression of the form   where   and   are the ends of the curve that you wish to integrate.

Integral calculusEdit

 
The integration region   is bounded by the curve   and the rays   and   .

Let   denote the region enclosed by a curve   and the rays   and   , where   . Then, the area of   is

 
 
The region R is approximated by n sectors (here, n = 5).

This result can be found as follows. First, the interval   is divided into   subintervals, where   is an arbitrary positive integer. Thus  , the length of each subinterval, is equal to   (the total length of the interval), divided by   , the number of subintervals. For each subinterval   , let   be the midpoint of the subinterval, and construct a circular sector with the center at the origin, radius   , central angle   , and arc length   . The area of each constructed sector is therefore equal to   . Hence, the total area of all of the sectors is

 

As the number of subintervals   is increased, the approximation of the area continues to improve. In the limit as   , the sum becomes the Riemann integral.

GeneralizationEdit

Using Cartesian coordinates, an infinitesimal area element can be calculated as   . The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered:

 

Hence, an area element in polar coordinates can be written as

 

Now, a function that is given in polar coordinates can be integrated as follows:

 

Here, R is the same region as above, namely, the region enclosed by a curve   and the rays   and   .

The formula for the area of   mentioned above is retrieved by taking   identically equal to 1.

ApplicationsEdit

Polar integration is often useful when the corresponding integral is either difficult or impossible to do with the Cartesian coordinates. For example, let's try to find the area of the closed unit circle. That is, the area of the region enclosed by   .

In CartesianEdit

 

In order to evaluate this, one usually uses trigonometric substitution. By setting   , we get both   and   .

   
 
 
 

Putting this back into the equation, we get

 

In PolarEdit

To integrate in polar coordinates, we first realize   and in order to include the whole circle,   and   .

 

An interesting exampleEdit

A less intuitive application of polar integration yields the Gaussian integral

 

Try it! (Hint: multiply   and   .)