Section 15.2

Iterated Integrals

Turning one hard double integral into two easy single integrals.

Introduction

Calculating Riemann sums is tedious. Just like the Fundamental Theorem of Calculus let us avoid summing rectangles in 1D, Iterated Integrals let us avoid summing boxes in 2D.

The strategy is "Partial Integration": Integrate with respect to while treating as constant (or vice versa).

Fubini's Theorem

The Theorem

If is continuous on the rectangle , then:

This means order does not matter for rectangles (as long as is continuous).

Visualizing Slices

Interactive: Cross-Sectional Area

is the area of the cross-section. Then we integrate along the y-axis.

Worked Examples

Example 1: Basic Polynomial

Evaluate .

1. Inner Integral (dy):

Treat x as constant.

2. Outer Integral (dx):

Example 2: Splittable Functions

Evaluate over .

Trick:

. Since limits are constants, we can split:
.

Execute:

Result: .

Example 3: Verifying Order

Show .

Order 1 (dy first):

Outer: .
.

Symmetry Check:

Since the function and limits are symmetric in x and y, the result must be the same!