Reverse order integration

In some situations it may be impossible to solve a multiple integral in one order (at least with finite terms), but possible in another. For example, in the double integral

\int 10 \int 1 \sqrt 𝑦 e x p (𝑥 3) 𝑑 𝑥 𝑑 𝑦

the inner integral cannot be solved¹. However, in reverse order, it is perfectly solvable.

\int 10 \int 1 \sqrt 𝑦 e x p (𝑥 3) 𝑑 𝑥 𝑑 𝑦 = \int 10 \int 𝑥 2 0 e x p (𝑥 3) 𝑑 𝑦 𝑑 𝑥 = \int 10 [𝑦 e x p (𝑥 3)] 𝑦 = 𝑥 2 𝑦 = 0 𝑑 𝑥 = \int 10 𝑥 2 e x p (𝑥 3) 𝑑 𝑥 = [13 e x p (𝑥 3)] 10 = 𝑒 - 1 3

Finding integral bounds

To reverse the order of an integral it is necessary to “reverse engineer” the integral bounds. Multiple coloured pens is recommended.

Practice problems

2016. Calculus, p. 1049 (§15.2 exercises 45–56)
2023. Advanced Mathematical Methods, p. 27 (§1 problems 3–4)

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2023. Advanced Mathematical Methods, p. 6 ↩

Reverse order integration

Finding integral bounds

Practice problems

Footnotes