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Calculus II: Vector Calculus and Multivariable Analysis / Multiple Integrals

Double Integrals over General Regions and Changing the Order of Integration

The joint density of two correlated log-normal asset prices is defined over R>02\mathbb{R}^2_{>0}, not a rectangle. Pricing an Asian option or computing the probability that both assets finish in-the-money requires integrating over a region bounded by curves — a half-plane, a wedge, or a payoff boundary. General-region double integrals are the mechanism for all such calculations. The crucial technique is changing the order of integration: because the limits of the inner integral depend on the outer variable, swapping dy dxdy\,dx to dx dydx\,dy can transform a problem with no closed-form inner antiderivative into a tractable one.
Learning objectives. By the end of this note you should be able to:
  • classify a planar region as Type I (vertical slices) or Type II (horizontal slices) and write the corresponding iterated integral;
  • switch the order of integration given a double integral with variable limits;
  • compute the area of a planar region using ∬D1 dA\iint_D 1\,dA;
  • set up joint probability calculations over triangular, trapezoidal, and curved regions.

In this lesson

    1. Type I and Type II Regions
    1. Setting Up the Iterated Integral
    1. Changing the Order of Integration
    1. Planar Area as a Double Integral

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