Iterated Integrals and Fubini's Theorem

A double integral generalizes the concept of a single-variable integral to functions of two variables, enabling the computation of the volume beneath a surface z = f(x, y) over a planar region R . For a rectangular region defined by a ≤ x ≤ b and c ≤ y ≤ d, and for functions continuous on this domain, the double integral can be evaluated as an iterated integral. This approach simplifies computation by reducing the problem to successive integrations with respect to one variable at a time.

Iterated Integration and Cross-sectional Area

Holding one variable constant during integration yields a cross-sectional area function. For instance, fixing x and integrating f(x, y) with respect to y over the interval [c, d] defines the function

\begin{equation*}A(x) = \int_{c}^{d} f(x,y)\, dy\end{equation*}

which represents the area of a vertical slice through the solid at position x. The total volume under the surface is then obtained by integrating this area along the x-axis:

\begin{equation*}\iint_{R} f(x,y)\, dA = \int_{a}^{b} A(x)\, dx = \int_{a}^{b} \int_{c}^{d} f(x,y)\, dy\, dx\end{equation*}

A similar argument applies when integrating first with respect to x and then with respect to y, using horizontal cross-sections.

Fubini’s Theorem

Fubini’s Theorem formalizes the validity of iterated integration. If f is continuous on a rectangular region R, then the double integral over R can be computed as:

\begin{equation*}\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\end{equation*}

This result ensures that the order of integration does not affect the value of the integral, providing flexibility in evaluating volumes and other quantities such as mass when density varies over a planar region.