Section 15.1: Double Integrals over rectangles

Recall that it is usually difficult to evaluate single integrals directly from the definition of an integral, but the Fundamental Theorem of Calculus provides a much easier method. The evaluation of double integrals from first principles is even more difficult, but here we see how to express a double integral as an iterated integral, which can then be evaluated by calculating two single integrals.

Suppose that \(f\) is a function of two variables that is integrable on the rectangle \(R = [a, b] \times [c, d]\). We use the notation \(\int_c^d f(x, y) dy\) to mean that \(x\) is held fixed and \(f(x, y)\) is integrated with respect to \(y\) from \(y = c\) to \(y = d\). This procedure is called partial integration with respect to y. Now \(\int_c^d f(x, y) dy\) is a number that depends on the value of \(x\), so it defines a function of \(x\): \[ A(x) = \int_c^d f(x, y) dy \] If we now integrate the function \(A\) with respect to \(x\) from \(x = a\) to \(x = b\), we get

\[ \tag{7} \int_a^b A(x) dx = \int_a^b \left[ \int_c^d f(x, y) dy \right] dx \]

The integral on the right side of Equation (7)) is called an iterated integral. Usually the brackets are omitted. Thus

\[ \tag{8} \int_a^b \int_c^d f(x, y) dy dx = \int_a^b \left[ \int_c^d f(x, y) dy \right] dx \] means that we first integrate with respect to \(y\) from \(c\) to \(d\) and then with respect to \(x\) from \(a\) to \(b\).

Similarly, the iterated integral \[ \tag{eq:9} \int_c^d \int_a^b f(x, y) dx dy = \int_c^d \left[ \int_a^b f(x, y) dx \right] dy \] means that we first integrate with respect to \(x\) (holding \(y\) fixed) from \(x = a\) to \(x = b\) and then we integrate the resulting function of \(y\) with respect to \(y\) from \(y = c\) to \(y = d\).

Notice that in both Equations (8) and (9) we work from the inside out.