I am stuck trying to solve the following integral:
$$\int_R (y+2x^2)(y-x^2) dA$$ where $R$ is defined by the following equations: $xy=1$, $xy=2$, $y=x^2$, $y=x^2-1$ with $x$ and $y$ positives.
I've tried several changes of variables for example: $u=xy$, $v=y-x^2$ or $u=y-x^2$, $v=x^2$ but I get stuck because for the Jacobian ot for the limits of integration I have to solve a third degree equation. I know that I could solve it using Cardano's formula but it has to be an easy way to do it.
Thank you very much. Merry Christmas.
As a first step it is better to represent the region of integration, as in the figure.
From this we see that, to integrate, we have to divide the domain in three subregions as:
1) $ x_A\le x\le x_B$ where we have $\frac{1}{x}\le y\le x^2$
2) $ x_B\le x\le x_C$ where we have $\frac{1}{x}\le y\le \frac{2}{x}$
3) $ x_C\le x\le x_D$ where we have $x^2-1\le y\le \frac{2}{x}$
This gives us the limits of integration and the integration is simple for the given function. The problem is to find the coordinates of the points $A,B,C,D$ where we find some difficulty in solving equations of third degree.
Can you do from there?