The risk-neutral pricing formula for a three-asset correlation swap, or the joint distribution of three factor returns, requires integrating a function against a trivariate density over a region in . Triple integrals are the natural extension of double integrals to three dimensions: the same Fubini strategy reduces the integral to three successive single-variable integrations, one variable at a time. The new challenge is that six different orders of integration are possible — and choosing the right order can make the difference between a tractable and an intractable calculation.
Learning objectives. By the end of this note you should be able to:
- set up a triple integral over a solid region as an iterated triple integral in any of the six possible orders;
- evaluate a triple integral by computing three successive single-variable integrals;
- use triple integrals to compute volumes of solids;
- identify the inner, middle, and outer limits when the domain is a tetrahedron, a pyramid, or a ball.
In this lesson
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- The Triple Integral
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- Integration over General Solids
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- Volume via Triple Integral
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- Choosing the Order of Integration
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- Triple Integrals for Probabilities
Before you start
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