Suppose R is the rectangle 1<=x<=4, |y|<=2 and evaluate the double integral ∫R∫f(x,y)dA, where f(x,y)= y/(1+3x^4)^(1/2).

Question

I first decided to integrate with respect to y first (which I think I can choose to do)

I am a little confused at this part though cause i would get zero for my answer.

So instead i though about it as the integral from 0 to 2 plus the integral from 0 to -2 with respect to y for both and what i got left is

1/2∫8/(1+3x^4)^(1/2)dx

at this part i thought about using trigonometric substitution but i am hesitant to go on further just cause i don't know if i'm thinking about this problem the right way.

Answer 1

The given function is odd in y. No matter how you look at it, the integral for a rectangular region from -a to +a will be zero.
If you split the integral into two, you should be adding
I(-2,0) to I(0,2) which is still zero.
So the answer is zero, and a quick one if it is a bonus question in the exam.