Hello, Looking for someone REALLY good at math!?

Question

Hello there! I'm looking for someone who can solve this double integral problem. Here is what is says:

"Evaluate the integral by making it an iterated integral" (or something similar to that)

Here is the problem:

int(0 to infinity) [ int(0 to infinity) xe^(-(x+2y))dxdy ]

Thank you so much in advance!

Answer 1

∫[x=0 to ∞]∫[y=0 to ∞] xe^(-x-2y) dx dy =

Here we separate the integrals, which we can do since the integrand is separable: it is in the form f(x)*g(y), so we can split the integral into two separate components.

(∫[y=0 to ∞] e^(-2y) dy ) * (∫[x=0 to ∞] xe^(-x) dx =
(-e^(-2y)/2 | y=0 to ∞]) * (-xe^(-x)-e^(-x) | [x=0 to ∞] ) =
(1/2) * (1) =
1/2

Answer 2

∫(0 to ∞) ∫(0 to ∞) xe^(-(x+2y)) dx dy
= ∫(0 to ∞) ∫(0 to ∞) xe^(-x).e^(-2y) dx dy
= ∫(0 to ∞) xe^(-x) dx . ∫(0 to ∞) e^(-2y) dy
= {[-xe^(-x)][0 to ∞] - ∫(0 to ∞) -e^(-x) dx} . [e^(-2y)/(-2)][0 to ∞]
= {0 - 0 - [e^(-x)][0 to ∞]} . (0 - 1/(-2))
= (0 + 1) (1/2)
= 1/2.