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,$$\displaystyle \int_R f(x,y) \ dxdy = \int_a^b\int_{g(x)}^{h(x)} f(x,y) dy dx$$

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.