Changing the Order of Integration
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Fubini’s Theorem
When discussing double integrals and iterated integrals, we learned about the following theorem:
Theorem: Let \(f\) be a function of two variables defined on a region \(R\) such that \(f\) and \(R\) are nice. Then the double integral of \(f\) over the region \(R\) and the two iterated integrals over the same region are equal:
\[\iint_R f(x,y)\;dA = \int_c^d \int_{a(y)}^{b(y)} f(x,y)\;dx\;dy = \int_a^b \int_{c(x)}^{d(x)} f(x,y)\;dy\;dx\]
So far we have used this theorem to evaluate double integrals by turning them into iterated integrals. We have also seen that for some regions \(R\), we can rewrite the double integrals as an iterated integral two ways, both \(dx\;dy\) and \(dy\;dx\). We also saw that depending on the function \(f\) and the region \(R\), one of these two iterated integrals is often much easier to calculate.
That can be used to calculate some difficult iterated integrals. Suppose you have an iterated integral that is hard to calculate. It is possible that by rewriting the iterated integral as a double integral, and then rewriting the double integral as an iterated integral with the opposite order of integration from the original iterated integral, the resulting iterated integral could be easier to calculate.
Examples of changing order of integration
Let’s look at some examples:
Example 1
Calculate \[\int_0^3\int_{x^2/3}^x xy\;dy\;dx\]
Example 2
Calculate \[\int_0^{2\sqrt{\pi}}\int_{y/2}^{\sqrt{\pi}} \cos\left(x^2\right)\;dx\;dy\]