Find double improper integral

Question

How do I evaluate the following case:

$f(x,y) = \begin{cases} \beta^2e^{-\beta y}, & 0 \leq x \leq y, & \text{where} & \beta > 0\\ 0 & \text{elsewhere} \end{cases}$

Find $\int_0^\infty \int_0^\infty f(x,y) dxdy$

I am very confused

Answer 1

We compute using integration by parts: \begin{align} \int_{[0,\infty)\times[0,\infty)}f(x,y)\ \mathsf d(x\times y) &= \int_0^\infty\int_0^y \beta^2 e^{-\beta y}\ \mathsf dx\ \mathsf dy\\ &= \int_0^\infty \beta^2 ye^{-\beta y}\ \mathsf dy\\ &=-\beta ye^{-\beta y}|_0^\infty + \int_0^\infty\beta e^{-\beta y}\ \mathsf dy\\ &= 0 - e^{-\beta y}|_0^\infty\\ &= 1. \end{align}