Home page for accesible maths 5 Chapter 5 contents 5.6 Solution to Example 5.5 5.8 Notation for double integrals

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5.7 Equality of repeated integrals

It is no co-incidence that these repeated integrals produce the same result.

Theorem (Fubini).

Let $f$ be continuous on a bounded rectangle $R$ . Then the repeated integrals are equal, and their common value is the double integral, so

\int\!\!\!\int_{R}f(x,y)\,dxdy=\int_{c}^{d}\Bigl\{\int_{a}^{b}f(x,y)\,dx\Bigr% \}dy=\int_{a}^{b}\Bigl\{\int_{c}^{d}f(x,y)\,dy\Bigr\}dx.

This result gives us a means of calculating double integrals by working out one-variable integrals via the fundamental theorem of calculus. In some cases, it can be much easier to calculate one of the repeated integrals than the other.