1 Attachment(s)

"Find the area outside the circle and inside the cardiod" - Double Integral Question

The question, solution and answer is attached. Could someone please explain to me how the 0, pi/2, 2, and 2(1+cos(theta)) boundaries are chosen/found and relate it to the drawing for me please?

Any input would be greatly appreciated!

Thanks in advance!

Re: "Find the area outside the circle and inside the cardiod" - Double Integral Quest

Can you find the area of the cardiod in the first quadrant?

Subtract the area of the circle in the first quadrant.

Double that number.

Re: "Find the area outside the circle and inside the cardiod" - Double Integral Quest

Okay, I looked a bit deeper into it and, assuming I haven't made any mistakes, rho is the radius (it would serve better to call it r but whatever, let's keep it the way it is) and, it is chosen as the lower limit of the inner integral to exclude the area of the circle (so I don't know why you asked me to subtract the circle's area from the first quadrant - were you trying to make the point that I need to exclude it's area in the calculation?).

All that now confuses me (if none of my assumptions so far are incorrect) is that the cardiod's area in the first quadrant is not like a larger circle in the first quadrant. More specifically, I don't see a "radial symmetry" (if such a term exists) from the angle passing through from the x axis to the y axis or, equivalently, from 0 to pi/2 so how can we just treat the cardiod's "radius" as a larger radius for the upper limit? If I am being unclear, please tell me.