Double integrals

A double integral $\iint_\Omega f(x,y)\,dA$ accumulates a quantity over a planar region $\Omega$ by adding contributions of the form $\text{density}\times\text{area}$. Depending on what $f$ represents, the result is a signed volume, a mass, a population or a probability. The chapter assumes one-variable integration and the multivariable function viewpoint.

Lessons

1. Double integrals and Riemann sums: defines the integral over a rectangle as a limit of sums and gives the upper-and-lower-sum test for integrability.
2. Fubini and iterated integrals: computes the integral on a rectangle by adding one-variable cross-sections, in either order.
3. Regions and order of integration: extends slicing to regions bounded by curves and shows how to reverse or split the order.
4. Change of variables and polar coordinates: reshapes the region or integrand, with the Jacobian determinant recording how area stretches.
5. Mass, centroids and probability: integrates a density to find mass, centres of mass and expected values.
6. Euler's Basel sum: combines a series, a linear change of variables and symmetry to prove $\sum 1/n^2=\pi^2/6$.
7. Seen questions: exam questions on integrability, upper sums and a change of variables on an elliptical region.

Where it leads

This last Calculus chapter extends the substitution rule $du=g'(x)\,dx$ from integration techniques to two variables; the absolute Jacobian determinant is also the area-scaling factor of a linear map's matrix in Algebra. Normalised densities turn the same integrals into continuous probability models.

Vocabulary checkpoints

• Double and iterated integrals: a double integral accumulates over a region; an iterated integral computes it by slicing.
• Order and region: changing the order changes the bounds, not the region.
• Jacobian matrix and determinant: in a coordinate change, the matrix is the local linear approximation and the absolute value of its determinant is the area-scaling factor.
• Density and integral: in mass and probability, a density is a rate per unit area; its integral is the total mass or probability.
• Region notation: $D$ usually denotes a function's domain, $R$ a rectangle and $\Omega$ the region of integration. $\partial\Omega$ is the boundary of $\Omega$, while $\partial$ applied to a function means partial differentiation.