Iterated Integrals | Sahithyan's Notes

An iterated integral evaluates a double integral one variable at a time.

For $f$ defined on $R = [a,b] \times [c,d]$ :

\int_a^b \int_c^d f(x,y)\;\text{d}y\,\text{d}x

The inner integral is computed first, treating $x$ as a constant. The result is a function of $x$ alone.

Fubini’s Theorem

If $f$ is continuous on $R = [a,b] \times [c,d]$ , then:

\iint_R f(x,y)\,\text{d}A = \int_a^b \int_c^d f(x,y)\;\text{d}y\,\text{d}x = \int_c^d \int_a^b f(x,y)\;\text{d}x\,\text{d}y

The order of integration can be swapped freely.

For non-rectangular regions, limits depend on variables. Swapping integration order requires re-expressing the region’s bounds.

For the region $\{(x, y) : a \leq x \leq b,\; g_1(x) \leq y \leq g_2(x)\}$ :

Find the full range of $y$ : $c \leq y \leq d$ .
For fixed $y$ , find the range of $x$ : $h_1(y) \leq x \leq h_2(y)$ .
Rewrite: $\displaystyle\int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y)\;\text{d}x\,\text{d}y$ .

For the triangular region $\{0 \leq y \leq 1,\; y \leq x \leq 1\}$ :

\int_0^1 \int_y^1 f(x,y)\;\text{d}x\,\text{d}y = \int_0^1 \int_0^x f(x,y)\;\text{d}y\,\text{d}x

Both integrals cover the triangle $\{0 \leq x \leq 1,\; 0 \leq y \leq x\}$ .

\int_0^1 \int_0^x y\;\text{d}y\,\text{d}x

Inner integral:

\int_0^x y\;\text{d}y = \left[\frac{y^2}{2}\right]_0^x = \frac{x^2}{2}

Outer integral:

\int_0^1 \frac{x^2}{2}\;\text{d}x = \left[\frac{x^3}{6}\right]_0^1 = \frac{1}{6}