Non rectangular regions of integration

Regions with verticalcross-sections

This type of region is limited by the interval $[a,b]$ in the variable $x$, and for certain functions $g(x)$, $h(x)$ in the variable $y$, or in other words $y\in [g(x),h(x)]$.

Then,$${\displaystyle {\int }_{R}f(x,y)dxdy={\int }_a^b{\int }_{g(x)}^{h(x)}f(x,y)dydx}$$

Regions with horizontal cross-sections

This type of region is limited by the interval $[c,d]$ in the variable $y$, and for certain functions $g(y)$, $h(y)$ in the variable $x$, or in other words $x\in [g(y),h(y)]$.

Then,$${\displaystyle {\int }_{R}f(x,y)dxdy={\int }_c^d{\int }_{g(y)}^{h(y)}f(x,y)dxdy}$$

Regions without cross-sections

In the case where the region does not have cross-sections, it is advisable to carry out a change of variable, imposing new variables, that will give us cross-sections.

On the other hand, it is possible that it will be necessary to separate the integral into different parts and that the integration limits have complex expressions.