Take
[tex]\begin{cases}u=x-y\\v=x+y\end{cases}[/tex]
so that
[tex]\begin{cases}\mathbf x(u,v)=\dfrac{u+v}2\\\\\mathbf y(u,v)=\dfrac{-u+v}2\end{cases}[/tex]
and the Jacobian determinant is
[tex]|\det J|=\left|\begin{vmatrix}\mathbf x_u&\mathbf x_v\\\mathbf y_u&\mathbf y_v\end{vmatrix}\right|=\dfrac12[/tex]
So the integral is (NOTE: I'm guessing on what the integrand is supposed to be)
[tex]\displaystyle\iint_R7xye^{x^2-y^2}\,\mathrm dA=\frac78\int_{u=0}^{u=10}\int_{v=0}^{v=4}e^{uv}(v^2-u^2)\,\mathrm dv\,\mathrm du[/tex]
[tex]=\dfrac{117453-5733e^{40}}{3200}[/tex]
