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Among the top uses of the 2dimensional changeofvariable formula are
The following video demonstrates ways to use the changeofvariable formula to simplify integrals. In it, we compute the integral of $e^{x+2y}$ over a parallelogram using a linear transformation, and the integral of $x^2$ over an offcenter ellipse using a rescaling and a shift. Linear transformations also appear in examples 2a and 2b, below. Example 1: Let's illustrate this change of variable idea in the case of polar coordinates. The Astrodome in Houston as shown to the right below might be modelled mathematically as the region below the cap of a sphere
The Astrodome problem showed how this works when ${\bf \Phi}$ is the change of coordinates from polar to Cartesian coordinates, but matrix mappings often help too. They usually come in two 'flavors' as indicated in
Solving 2a should be easier than 2b because we are already given the transformation, but both cases are very similar in the sense that we'll end up solving a pair of simultaneous equations.
