The integral of a function over a region in the plane or in space underpins the most important formulas in quantitative finance: the expected value of a random variable with a multivariate density, the price of a basket option computed by integrating a joint payoff against a correlated lognormal density, and the Black-Scholes formula itself — which can be read as a Gaussian integral over the risk-neutral measure. This topic extends single-variable integration to double integrals over planar regions and triple integrals over solid regions, introduces the all-important technique of changing variables via the Jacobian determinant, and builds up to the full suite of coordinate systems (polar, cylindrical, spherical) that make otherwise intractable integrals computable.
By the end of this topic you should be able to:
- define the double integral as a limit of Riemann sums and evaluate it as an iterated integral using Fubini's theorem;
- set up double integrals over general (non-rectangular) regions by choosing the correct order of integration;
- compute volumes and probabilities by integrating over planar and solid regions;
- apply the change-of-variables formula with the Jacobian determinant, including the polar and spherical cases;
- evaluate triple integrals in Cartesian, cylindrical, and spherical coordinates;
- compute mass, centre of mass, and moments of inertia via multiple integrals;
- recognise the Gaussian integral as the prototype for the normal density normalisation.
In this lesson
- Contents
- The big picture
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