Why change the order?
Before you start: read double integrals first.
Some double integrals are impossible in the order they're written. For example, \(\displaystyle\int e^{y^2}\,dy\) has no formula in terms of elementary functions. Integrating over the same region in the other order can make the problem easy.
1. Read the region from the given limits
Write the inequalities, for example \(0 \le x \le 1\), \(x \le y \le 1\).
2. Sketch it
Draw the boundary curves and shade the region.
3. Slice it the other way
Draw a strip in the other direction and read the new limits. The outer limits must be numbers.
The idea in one picture
Solved examples
Example 1: An impossible integral made easy
MediumEvaluate \(\displaystyle\int_0^1\!\!\int_x^1 e^{y^2}\,dy\,dx\).
- Region: \(0 \le x \le 1\), \(x \le y \le 1\), the triangle in the picture.
- With horizontal strips: \(y\) from 0 to 1, and \(x\) from 0 to \(y\). So the integral is \(\displaystyle\int_0^1\!\!\int_0^y e^{y^2}\,dx\,dy\).
- Inner: \(y\,e^{y^2}\). Outer: \(\displaystyle\int_0^1 y\,e^{y^2}\,dy = \Big[\tfrac{1}{2}e^{y^2}\Big]_0^1\).
Answer\(\dfrac{e - 1}{2}\)
Example 2: Another one
MediumEvaluate \(\displaystyle\int_0^1\!\!\int_x^1 \sin(y^2)\,dy\,dx\).
- Same region as Example 1, so it becomes \(\displaystyle\int_0^1\!\!\int_0^y \sin(y^2)\,dx\,dy = \int_0^1 y\sin(y^2)\,dy\).
Answer\(\dfrac{1 - \cos 1}{2}\)
Example 3: Between a parabola and a line
MediumChange the order of \(\displaystyle\int_0^1\!\!\int_{x^2}^{x} f(x, y)\,dy\,dx\).
- Region: \(x^2 \le y \le x\), \(0 \le x \le 1\).
- Solve each curve for \(x\): \(y = x\) gives \(x = y\), and \(y = x^2\) gives \(x = \sqrt y\). For a horizontal strip, \(x\) runs from \(y\) (left) to \(\sqrt y\) (right).
Answer\(\displaystyle\int_0^1\!\!\int_{y}^{\sqrt y} f(x, y)\,dx\,dy\)
Example 4: Infinite limits
Exam levelEvaluate \(\displaystyle\int_0^\infty\!\!\int_x^\infty \frac{e^{-y}}{y}\,dy\,dx\).
- Region: \(0 \le x \le y\), with \(y\) going to \(\infty\).
- Reversed: \(\displaystyle\int_0^\infty\!\!\int_0^y \frac{e^{-y}}{y}\,dx\,dy = \int_0^\infty \frac{e^{-y}}{y}\cdot y\,dy\).
Answer\(\displaystyle\int_0^\infty e^{-y}\,dy = 1\)
Example 5: A square root
Exam levelEvaluate \(\displaystyle\int_0^1\!\!\int_{\sqrt y}^{1}\sqrt{1 + x^3}\,dx\,dy\).
- Region: \(\sqrt y \le x \le 1\), \(0 \le y \le 1\), that is, \(0 \le y \le x^2\), \(0 \le x \le 1\).
- Reversed: \(\displaystyle\int_0^1\!\!\int_0^{x^2}\sqrt{1 + x^3}\,dy\,dx = \int_0^1 x^2\sqrt{1 + x^3}\,dx\).
- Substitute \(u = 1 + x^3\): \(\dfrac{2}{9}\Big[(1 + x^3)^{3/2}\Big]_0^1\).
Answer\(\dfrac{2}{9}\big(2\sqrt 2 - 1\big)\)
If the inner integrand has no elementary antiderivative, such as \(e^{y^2}\), \(\sin y^2\) or \(\dfrac{\sin y}{y}\), that's the examiner's signal to change the order.
Common mistakes
1. Just swapping the limits
\(\int_0^1\!\int_x^1 \dots\,dy\,dx\) does not become \(\int_x^1\!\int_0^1 \dots\,dx\,dy\). The new limits must come from the sketch.
2. Left and right mixed up
For a horizontal strip, the lower limit is the curve on the left, the upper limit the curve on the right.
3. Missing a split
If the left or right boundary changes part way up the region, split it into two integrals.
Practice questions
Q1\(\displaystyle\int_0^1\!\!\int_y^1 e^{x^2}\,dx\,dy\)
Reverse: \(\displaystyle\int_0^1\!\!\int_0^x e^{x^2}\,dy\,dx = \int_0^1 x e^{x^2}\,dx = \frac{e - 1}{2}\).
Q2\(\displaystyle\int_0^2\!\!\int_x^2 e^{-y^2}\,dy\,dx\)
Reverse: \(\displaystyle\int_0^2 y e^{-y^2}\,dy = \frac{1 - e^{-4}}{2}\).
Q3\(\displaystyle\int_0^\pi\!\!\int_x^\pi \frac{\sin y}{y}\,dy\,dx\)
Reverse: \(\displaystyle\int_0^\pi \frac{\sin y}{y}\cdot y\,dy = \int_0^\pi\sin y\,dy = 2\).
Q4Change the order of \(\displaystyle\int_0^2\!\!\int_{x^2}^{2x} f\,dy\,dx\).
The curves meet at \(x = 0, 2\), with \(y\) from 0 to 4. Horizontal strip: \(x\) from \(\dfrac{y}{2}\) to \(\sqrt y\). So \(\displaystyle\int_0^4\!\!\int_{y/2}^{\sqrt y} f\,dx\,dy\).
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Frequently asked questions
When should I change the order of integration?
When the inner integral can't be done in elementary terms, or when the other order needs fewer pieces.
Does changing the order change the answer?
No. It is the same integral over the same region; only the way you slice the region changes.
Where this leads next
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Vipul Sir has taught mathematics for over 15 years to HSC, CBSE, ICSE, IGCSE/IB, Diploma, Engineering and BSc students at VVS Classes in Goregaon and Vile Parle, Mumbai. He teaches every class himself, concepts first and exam technique second.