Double integral And polar coordinate system

Question

I have to evaluate this integral over the domain D

The Plot would be like this:

I decided to use polar coordinate system using it

It gives me this but I don't know the upper limit of integration

Help me finish this example and state my probable mistakes. Please !

Thanks

Answer 1

As a double integral, you will be integrating out to a radius given by the polar curve $ \ r \ = \ 2 \cos \theta \ $ , over the interval in angle $ \ \theta \ , [ \ 0 \ , \ \frac{\pi}{2}] \ $ . This is because the "circle" of which your semi-circular domain is a part is the "one-petal" rosette, $ \ r \ = \ 2 \cos \theta \ $ . The semi-circle lies only in the first quadrant, so $ \ \theta \ $ only runs over the interval mentioned.

Note that this means you cannot "separate" the variable integrations.