Limit of integration - double integral

Question

$$\iint_D f $$

where $D$ is region given by $D = \{(x,y) | 0 < ax + by < \pi,\ 0< cx + dy < \pi\}$

My question is how can i find the limit of integration, is it by change of variables? Please only give me hints about limits of integration and please also do not evaluate the question

Thank you very much

Answer 1

Make a change of variables $$ u = ax + by, \ v = cx + dy $$

Then

$$ du dv = \left|\begin{matrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \end{matrix}\right| dxdy = (ad - bc)\ dxdy $$

Thus

$$ \iint_{D(x,y)} \sin(ax+by)\sin(cx+dy)\ dxdy = \frac{1}{ad-bc}\iint_{D(u,v)} \sin(u)\sin(v)\ dudv $$

The limits of integration should be obvious now.

Answer 2

More generally: when the domain of the double integral $$\iint_D f$$ is limited by level curves of nice functions: $$D = \{(x,y)\,|\, g_1\le g(x,y)\le g_2, h_1\le h(x,y)\le h_2\}$$ and the transformation $$(u,v) = (g(x,y),h(x,y))$$ is injective, we have usually a good change of variables.

Ditto for triple integrals with domains limited by level surfaces.