I'm new to this forum and trying to self-study Calc 3 from an online textbook. I'm working through the practice-sets of this book, but the solution appears to be different from what I had expected.
This is a link to the problem. I had converted the bounds of the two integrals successfully, but when met with the equation within, I had assumed that it would be $y = r*sin(\theta)$, like in standard 2D rectangular to polar conversion methods. Instead, the correct solution appears to be $r^2*sin(\theta)$, and I'm not sure where this additional multiplicand of r comes from, or if I'm approaching this correctly at all. I'd appreciate any help.
Thank you!
When some change of variables is done you also have to multiply by the jacobian of the change of variables which, in the case of polar coordinates, is the radius $r$
https://en.m.wikipedia.org/wiki/Jacobian_matrix_and_determinant