Fubini's Theorem

The following theorem gives a practical method for evaluating a double integral by expressing it as an iterated integral (in either order).

10 Fubini’s Theorem If \(f\) is continuous on the rectangle \(R = \{(x, y) | a \le x \le b, c \le y \le d\}\), then \[ \iint_R f(x, y) dA = \int_a^b \int_c^d f(x, y) dy dx = \int_c^d \int_a^b f(x, y) dx dy \] More generally, this is true if we assume that \(f\) is bounded on \(R\), \(f\) is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.