Change of Variables in Multiple Integrals

In one-dimensional calculus we often use a change of variable (a substitution) to simplify an integral. By reversing the roles of x and u, we can write the Substitution Rule (5.5.6) as

\[ \int_a^b f(x) dx = \int_c^d f(g(u)) g'(u) du \tag{1} \]

where \(x = g(u)\) and \(a = g(c)\), \(b = g(d)\). Another way of writing Formula 1 is as follows:

\[ \int_a^b f(x) dx = \int_c^d f(x(u)) \frac{dx}{du} du \tag{2} \]

A change of variables can also be useful in double integrals. We have already seen one example of this: conversion to polar coordinates. The new variables r and \(\theta\) are related to the old variables x and y by the equations \[ x = r \cos \theta \qquad y = r \sin \theta \] and the change of variables formula (15.3.2) can be written as \[ \iint_R f(x, y) dA = \iint_S f(r\cos\theta, r\sin\theta) r dr d\theta \] where S is the region in the \(r\theta\)-plane that corresponds to the region R in the xy-plane.