Question

Use the given transformation to evaluate the integral.

where R is the region bounded by the ellipse

Answer 1

Step 1 :

Change of variable in double integral :

Conversion of integral from rectangular to polar co-ordinates :

Step 2 :

The integral is and the change of variables .

Where R  is the region bounded by the ellipse .

Substitute and in above equation.

Since the domain is a unit circle, the limits of integral are and

Find the Jacobian :

 

Step 3 :

 Evaluate the integral :

Write the above integral in polar co-ordinates :

Substitute and .

Since the domain is a unit circle, the new limits are and .

Solution :