The right coordinates for … are … and …, with … and…

R is a pie-shaped wedge: $0⩽r⩽1$ and $\pi /4⩽\theta ⩽3\pi /4$.

The right coordinates for $R^{\ast }$ are $r^{\ast }$ and ${\theta }^{\ast }$, with $x=r^{\ast }\cos {\theta }^{\ast }$ and $y=r^{\ast }\sin {\theta }^{\ast }+1$.
(a) Show that $J=r^{\ast }$ so $dA=r^{\ast }d{r}^{\ast }d{\theta }^{\ast }$.
(b) Find limits on $\iint r^{\ast }d{r}^{\ast }d{\theta }^{\ast }$ to integrate over $R^{\ast }$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §14.2 Change to Better Coordinates, #9