I am studying how we use polar substitution to solve double integrals. However, I am struggling with finding the correct limits of the transformed integrals to obtain a suitable solution. eg:
Why do we integrate between 0 and 1, then 0 and $2*\pi$?
Thanks
Note that $x^2 + y^2 = 1$ is the unit circle (radius $1$). So $x^2 + y^2 \leq 1$ describes the unit disk (the circle and its interior).
To obtain the area of the unit disk, using polar coordinates, $x^2 + y^2 = r^2 \leq 1 \implies 0\leq r \leq 1$, (the points in the unit disk are of distance $d$ which ranges from $0$ to $1$ units from the origin), and $\theta$ ranges from $0$ to $2\pi$ (one full revolution needed to reach every point).