# Double integrals of modulo x + modulo y

Is the following statement:

$\int_{x_1}^{x_2}{\int_{y_1}^{y_2}{g(x)+h(y)dxdy}} = \int_{x_1}^{x_2}{g(x)dx} + \int_{y_1}^{y_2}{h(y)dy}$

False in the general case ? Does it hold if $g(x) =|x|$ and $h(y) = |y|$ ?

\begin{align} \int_{x_1}^{x_2}\int_{y_1}^{y_2}g(x)+h(y)\ dxdy &= \int_{x_1}^{x_2}\int_{y_1}^{y_2}g(x)\ dxdy + \int_{x_1}^{x_2}\int_{y_1}^{y_2} h(y)\ dxdy \\ &= (y_2-y_1)\int_{x_1}^{x_2}g(x)\ dx + (x_2-x_1)\int_{y_1}^{y_2}h(y)\ dy \end{align}