## Calculus: Early Transcendentals (2nd Edition)

$\int_{0}^{2}\int_{1}^{3} xy$ $dy dx$ or $\int_{1}^{3}\int_{0}^{2} xy$ $dx dy$
We have to integrate with respect to $x$, where $0\leq x\leq 2$ and integrate with respect to $y$, where $1 \leq y \leq 3$. To do this, we use a double integral, where if we decide to integrate with respect to $y$ first, we needed to use $dydx$ and use $dxdy$ if we decide to integrate with respect to $x$ first. Regardless, the integral bounds must reflect the variable being integrated over. For instance, for $\int_{0}^{2}\int_{1}^{3} xy$ $dy dx$ since we are integrating with respect to $dy$ first, the bound of the first integral must be $1$ to $3$.