## Change of variable exercise

Here is an integral: \begin{align*} Q &= \int_0^1 \int_0^1 x^2 y \; dx dy \\ &= 1/6 \end{align*} Easy enough, but what if we change the variables like this: \begin{align*} u &= x \\ v &= xy \end{align*} then \begin{align*} J &= \begin{pmatrix} 1 & 0 \\ y & x \end{pmatrix} \\ dudv &= |\det(J)| \; dxdy \\ &= x\; dxdy \\ dxdy &= (1/x) dudv \end{align*} Substitute everything into $Q$: \begin{align*} Q &= \iint_? uv (1/x) \; dudv \\ &= \iint_? v \; dudv \end{align*} The only thing we need now is the boundary. We acquire this by mapping everying point $(x, y)$ to $(u, v)$ and transform the 1x1 square area into a triangle (see the figure). And from there we have \begin{align*} Q &= \int_0^1 \int_0^u v \; dvdu \\ &= 1/6 \end{align*}