How to Solve Double Integrals by Using Polar Coordinates – Example 2

In this mathematics and calculus tutorial, we explain how to solve double integrals by using polar coordinates. The YouTube tutorial is given below.

The problem of Solving Double Integrals

Calculate the value of the double integral

(1)  

where the domain of integration is the circle with the radius of 1.

Solve Double Integrals by Using Polar Coordinates

To solve this integral, we introduce the polar coordinates

(2)  

The integration domain is a circle with the radius of . The equation of the circle boundary in the Cartesian coordinates is

(3)  

The inside of the circle is described by the following inequality

(4)  

Consequently, in the Cartesian coordinates, the integration domain is

(5)  

By substituting the polar coordinates (6) in

(6)  

We obtain

(7)  

The angle is in the interval from to since we need to describe the complete circle. Consequently, the integration domain in the polar coordinates takes the following form

(8)  

On the other hand, the term in the original integral (1) can be expressed as follows in the polar coordinates

(9)  

By substituting (6) and (9) in the original integral (1) and by taking into account the integration domain in the polar form (8), the original integral becomes:

(10)