This is a follow-up question to Integrating multivariate piecewise function where one condition involves multiple variables
In the case where a condition depends on another condition, how would you go about evaluating the integral? For example, $\int_{-\infty}^\infty f(x,y) dy$ where:
$$f(x,y) = \begin{cases}
g(x,y) & -y \leq x \leq y, 0 < y < \infty \\
0 & \text{otherwise}
\end{cases}$$
Best Answer
This answer was given to me by a teaching assistant from my home university.
It would help to first visualize the domain over where $f(x, y)$ is nonzero (or, in this case, equivalently, where $f(x, y) = g(x, y)$) by sketching a diagram. Then it follows from the diagram that
$$\int_{-\infty}^{\infty} f(x, y) dy = \begin{cases} \int_{-x}^{\infty} g(x, y) dy & x < 0 \\ \int_x^{\infty} g(x, y) dy & x \geq 0 \end{cases}$$