Changing variables is a very useful technique for simplifying many types of math problems. You can use a horizontal translation $f(x) \, \to\, f(x+1)$ to change a parabola $y = f(x) = x^2 - 2x + 1$ with vertex at the $(1,0)$ into another parabola $y = f(x+1) = x^2$ with vertex at the origin. We also use changes of variables to convert hard integrals into easier integrals. The change-of-variables formula for ordinary integrals is $$\int_a^b\, f(x)\, dx \ = \ \int_{\alpha}^{\beta}\, f\big(g(u)\big)g'(u)\, du\,, \qquad g : [\alpha,\, \beta] \ \to \ [ a,\,b]\,.$$

Transformations in higher dimensions, called maps or mappings, play an even more important role in multi-variable calculus. We have already seen one such mapping ${\bf \Phi} : {\mathbb R}^2\,\to\, {\mathbb R}^2$, namely polar coordinates: $${\bf \Phi}: (r,\, \theta)\, \to\, (x,\, y)\,, \qquad x \ = \ r\cos \theta\,, \quad y \ = \ r\sin \theta\,,$$

The reason mappings like these are so useful in double integrals comes from their action on particular sets in the plane.

  A mapping ${\bf \Phi} : {\mathbb R}^2\,\to\, {\mathbb R}^2$ and a rectangle $D^*$ with sides parallel to the axes in the $uv$-plane such that: $${\bf \Phi}(u,\, v) \ = \ (x(u,\,v),\, y(u,\,v))\,, \qquad {\bf \Phi}\big(D^*\big) \ = \ D\,;$$

  A 'distortion' function $\displaystyle{\frac{\partial(x,\,y)}{\partial(u,\,v)}}$ to replace $g'(u)$ so that $$\int\int_D \, f(x,\, y)\, dxdy \ = \ \int\int_{D^*}\, f\big({\bf \Phi}(u,\,v)\big)\, \left|\frac{\partial(x,\,y)}{\partial(u,\,v)}\right|\,dudv\,.$$

In this case, if $D^* = [a,\,b]\times[c,\,d]$, then $$\int\int_D \, f(x,\, y)\, dxdy \ = \ \int_a^b \left(\int_c^d \, f\big({\bf \Phi}(u,\,v)\big)\, \left|\frac{\partial(x,\,y)}{\partial(u,\,v)}\right|\,dv\right)du\,.$$

When the region of integration $D$ in the $xy$ plane has rotational symmetry, polar coordinates often send a rectangle $D^*$ in the $r\,\theta$ plane to a more complicated region $D$.