Changes of variables in double integrals

There are many cases in which the variables $x$ and $y$ will not be useful for solving the integral, maybe because of the function we want to integrate or because the interval in which we integrate is too complicated.

In these cases, we will look for changes of variables that simplify the resolution to the problem.

The change of variable with $2$ variables is done in a similar way as with one variable, as in the following procedure:

• Given the initial variables $x$ and $y$, choose the functions $u(x, y)$ and $v(x, y)$, the new variables of the system, which preferably give us an easier domain of integration.

• Compute the Jacobian matrix of the ancient coordinates regarding the new variables:$\displaystyle \begin{bmatrix} \frac{dx}{du} & \frac{dx}{dv}o \\ \frac{dy}{du} & \frac{dy}{dv} \end{bmatrix}$, and compute the absolute value of its determinant $\displaystyle |J|=ABS\left(\left|\begin{bmatrix} \frac{dx}{du} & \frac{dx}{dv} \\ \frac{dy}{du} & \frac{dy}{dv}\end{bmatrix}\right|\right)$

• Compute the integration domain in the new variables $u$, $v$ (which we will call $\widehat {R}$ ) either form the analytical expression or the drawing of the region. Also compute the function $f (x, y)$ according to $u$, $v$. We will call it $\widehat{f}(u,v)$.

• $\displaystyle \int_R f(x,y) \ dxdy=\int_{\widehat{R}} \widehat{f}(u,v)\cdot |J| \ dudv$