Iterated Integrals
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 5.1: “Double Integrals over Rectangular Regions” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/5-1-double-integrals-over-rectangular-regions |
| Textbook used in class | Stewart, Calculus, Section 15.2: “Iterated Integrals” (Examples 1, 2, 3) |
Quick Reference
Fubini’s Theorem: For $f$ continuous on $R = [a,b]\times[c,d]$: $$\iint_R f(x,y)\,dA = \int_a^b\int_c^d f(x,y)\,dy\,dx = \int_c^d\int_a^b f(x,y)\,dx\,dy.$$
Procedure: In $\int_a^b\int_c^d f(x,y)\,dy\,dx$, evaluate the inner integral first by treating $x$ as a constant.
Motivation
The definition of the double integral as a limit of Riemann sums tells you what the integral means, but provides no direct method to compute it. Fubini’s theorem is the computational tool: it says you can evaluate a double integral by doing two single-variable integrations in sequence, one at a time. The order does not matter (for continuous $f$ on a rectangle).
Key Concept
An iterated integral $\int_a^b\int_c^d f(x,y)\,dy\,dx$ is computed in two steps:
Step 1 (inner integral): Hold $x$ fixed and integrate $f(x,y)$ with respect to $y$ from $c$ to $d$. This produces a function of $x$ only.
Step 2 (outer integral): Integrate the result from Step 1 with respect to $x$ from $a$ to $b$.
Reading the order of $dy\,dx$: integrate with respect to $y$ first (inner), then $x$ (outer).
Worked Examples
Example 1. Evaluate $\int_0^3\int_1^2 x^2 y\,dy\,dx$. (Stewart 15.2, Example 1.)
Inner integral (integrate w.r.t. $y$, $x$ is constant): $$\int_1^2 x^2 y\,dy = x^2 \left[\frac{y^2}{2}\right]_1^2 = x^2\left(\frac{4}{2} - \frac{1}{2}\right) = \frac{3x^2}{2}.$$
Outer integral: $$\int_0^3 \frac{3x^2}{2}\,dx = \frac{3}{2}\left[\frac{x^3}{3}\right]_0^3 = \frac{3}{2}\cdot 9 = \frac{27}{2}.$$
Example 2. Evaluate $\int_0^{\pi/2}\int_0^1 y\sin(xy)\,dx\,dy$.
Inner integral (w.r.t. $x$, $y$ is constant): $$\int_0^1 y\sin(xy)\,dx = y\cdot\left[-\frac{\cos(xy)}{y}\right]_0^1 = [-\cos(xy)]_0^1 = -\cos y + \cos 0 = 1 - \cos y.$$
Outer integral: $$\int_0^{\pi/2}(1-\cos y)\,dy = \left[y - \sin y\right]_0^{\pi/2} = \frac{\pi}{2} - 1.$$
in $\int_a^b\int_c^d f\,dy\,dx$, integrate with respect to $x$ first. The notation $dy\,dx$ means integrate with respect to $y$ first (the inner $dy$), then with respect to $x$ (the outer $dx$). Think of it as reading the differentials from left to right: the first one you encounter is the inner integral. This is the OPPOSITE of the typical order of the limits (the outer limits $a$ to $b$ come first in the notation but correspond to the outer, final integral).
Leveled Practice
Problem 1. Evaluate $\int_0^2\int_0^1 (x + e^y)\,dx\,dy$.
Show answer
Inner (w.r.t. $x$): $\int_0^1(x+e^y)\,dx = [x^2/2 + xe^y]_0^1 = 1/2 + e^y$.
Outer (w.r.t. $y$): $\int_0^2(1/2 + e^y)\,dy = [y/2 + e^y]_0^2 = (1+e^2) - (0+1) = e^2$.