Integrating an exponential function over [itex]|x|+|y| \leq 1[/itex]

[math.geek]

OK, I'm new to multi-variable calculus and I got this question in my exercises that asks me to integrate e^{-2(x+y)} over a diamond that is centered around the origin:

\int\int_D e^{-2x-2y} dA

where D=\{ (x,y): |x|+|y| \leq 1 \}

I know that the region I'm integrating over is symmetric over the x-axis and the y-axis, but e^{-x} or e^{-y} are neither odd nor even to use the symmetry that way.

Obviously, the diamond is symmetric over the axes x+y and x-y. Does this help?

 

[ShayanJ]

Split it into two integrals where one is over one half of the diamond and the other is over the other half!
Then you can write the boundaries of the integrals from D=\{ (x,y): |x|+|y| \leq 1 \} easily.

 

[math.geek]

Which haves?

Give me some more details please, I'm not looking for a full solution, only an explanation of how you're setting up the integral.

 

[ShayanJ]

One choice is dividing the diamond in half by the y axis.Then you have two double integrals,one for the left half and the other for the right one.The interval for the left half integral is -(x+1) \ to \ x+1 for integration w.r.t. y and -1 to 0 for integration w.r.t. x.I think you can figure out the intervals for the right half integral by plotting D.