Multiple integrals and iterated integration

For a continuous function on a closed rectangle, the double integral can be evaluated by integrating one coordinate first and then the other. The inner integral treats the outer variable as fixed. On a nonrectangular region, each slice can have different endpoints. A vertically simple region is described by x between fixed endpoints and y between two functions of x. The integral then uses exactly those moving y bounds. Reversing order requires re-describing the same set using horizontal slices, rather than merely swapping the symbols dx and dy.

Drawing a rough boundary sketch and recording inequalities prevents many mistakes. For a triangle, a horizontal slice may begin at a sloping edge and end at a vertical one, whereas a vertical slice may begin at a horizontal edge. More complicated regions may need splitting because one formula cannot describe every slice. Boundaries of ordinary piecewise smooth planar regions have area zero, so including or excluding those boundaries does not affect an integrable area integral. This observation does not permit discarding a positive-area piece or ignoring an interior singularity.

Tonelli's theorem applies to a nonnegative measurable function on a product of sigma-finite measure spaces. Both iterated integrals equal the product integral, allowing the common value to be infinite. It is therefore an effective first tool for testing absolute integrability: apply it to the absolute value of a signed function. Fubini's theorem then guarantees that an absolutely integrable function has integrable slices almost everywhere and equal finite iterated integrals. Exceptional slices of measure zero do not invalidate that conclusion.

A conditional cancellation is not the same as absolute integrability. It can hide dependence on the order of limiting operations near a singularity or at infinity. If a signed integral is not absolutely integrable, computing two iterated expressions and seeing different answers signals that the standard Fubini guarantee was unavailable. One should specify the intended improper-limit convention or use another justified theorem, rather than declaring that integration order never matters. On compact ordinary regions, a continuous integrand avoids these issues because it is bounded and absolutely integrable.

Changing order is often the difference between a difficult antiderivative and an elementary one. For example, if an integrand depends only on y, integrating first in x can turn the problem into the integrand times the horizontal width of the region. This is not a trick: it is a geometric evaluation of equal slices. Conversely, a density concentrated near a curved boundary might be simpler after a coordinate change, which is a different operation requiring an area-scale factor and will be treated separately.

Multiple integrals retain units. A mass density in kilograms per square meter multiplied by area produces mass; integrating only over one coordinate gives a line density until the second integration is performed. Integrating the constant one gives region area, making it a useful check on bounds. Symmetry is another check: an integrable function odd in x on a region symmetric under x reversal has zero integral. Before relying on symmetry, verify that the transformation preserves the region and that the signed integral actually exists.