Evaluating a double integral from zero to infinity

Question

How do I evaluate this integral? I don't understand at which point the limit notation should set in? And my method yields $0$ in the end. The integral is: $$ \int_0^{\infty} \int_0^{\infty} c\,x\,y\,e^{-(x+y)} \;\mathrm{d}y\;\mathrm{d}x $$

Answer 1

You can separate $x$ and $y$ and integrate successively with respect to each variable since $$cxye^{-(x+y)}=c(xe^{-x})(ye^{-y})$$ Now $$\int_{0}^{\infty}xe^{-x}dx$$ is equal to the expectation of an exponentially distributed random variable with parameter $1$. Therefore it is also equal to $1$. Thus $$ \int_0^{\infty} \int_0^{\infty} cxye^{-(x+y)} \;\mathrm{d}y\mathrm{d}x =c \int_0^{\infty}ye^{-y} \left(\int_0^{\infty}xe^{-x} \;\mathrm{d}x\right)\mathrm{d}y=c\cdot1\cdot1=c $$

Answer 2

Hint: Your integral is basically product of two gamma functions. $$\int_0^{\infty} \int_0^{\infty} c\,x\,y\,e^{-(x+y)} \;\mathrm{d}y\;\mathrm{d}x=c\int_0^{\infty} x\,e^{-x} \;\mathrm{d}x\int_0^{\infty} y\,e^{-y} \;\mathrm{d}y$$