# Thread: Double integral of a function on a region delimited by quadratic functions

1. ## Double integral of a function on a region delimited by quadratic functions

Hi,

I'm looking for the steps to solve this kind of questions.

Calculate the integral $\displaystyle \int \int_D \! (xy)^2 \, \mathrm{d}A$ where D is the region delimited by the following functions and relations: $\displaystyle y=4-x^2, y=x^2-4, x=1+y^2, x=-1-y^2$

When I try to solve it, I never find a way to a numeric answer. I always get a big function which I'm sure it's not.

Thank you so much for giving me a hint,

Bazinga

2. ## Double integral of a function on a region delimited by quadratic functions

Hi. Because the solution had a picture, I wrote it on piece of paper! Sorry for my bad hand writing.

You answer was really helpful, thanks a lot! It helped me visualize the problem. I think I found a way to solve it but I'm not quite sure. I try to follow your solution but I don't know how you build the x^4 equation.

Anyway, I came out with a solution. I'm not so sure about it but I'm posting it if anybody can tell me it's a good or bad reasoning.

Thanks,

Bazinga

4. ## Double integral of a function on a region delimited by quadratic functions

Well about building $\displaystyle x^4$ equation. It's simple! If we want to find the coordinates of "point" we should put these two equations equal to each other.
$\displaystyle y=4-x^2$ and $\displaystyle x=1+y^2$ then we get $\displaystyle x^4-8x^2-x+17=0$. I can solve this equation with maple, but I would like to solve it in a classic way! The answers are x=1.767469062 and x=2.263652586. But we only need the first one. I used x=1.77 for simplicity. Then y=0.87.

My answer is "33.83" which is equal to yours, but I solved it on my own at first and then checked it with maple!

To solve the integrals we should use this formula: $\displaystyle \int udv= uv - \int vdu$.