Changing the Order of Integration
Changing the order of integration sometimes leads to integrals that are more easily evaluated; Conversely, leaving the order alone might result in integrals that are difficult or impossible to integrate.
You can change the order of integration—and get the same result— if the limits of all variables are constant. However, if you have variable limits of integration and you change the order of integration, you must also change the limits of integration (Ram, 2009). In visual terms, let’s say you have a double integral that involves a vertical strip, sliding along the x-axis. If you change the order, you then have a horizontal strip sliding along the y-axis.
Example question: Compute the following double integral:
A quick glance at this integral reveals a problem: The “inside” integral (integrating with respect to x) requires you to find the antiderivative of 1/√(x3 + 1). There isn’t an integration rule that can help with that, so we’re going to switch the order of integration to find a solution.
Step 1: Write the limits of integration as inequalities:
- (0 ≤ y ≤ 1 )
- (√y ≤ x ≤ 1 )
Step 2: Find a new set of inequalities that describes the region with the variables in opposite order. Note: This step is much easier if you draw a graph of the area.
In the set of inequalities from Step 1, y came first. So first, define the region in terms of x instead.
A quick look at the graph tells you that the area is bounded from the left and right by x-values ranging from 0 to 1, so we have:
(0 ≤ x ≤ 1)
Next, you’ll want to define the shape in terms of the y-variable; This is the area bounded from above and below.
The area is bounded by the line y = 0 (i.e. the x-axis) on the bottom and the equation y = x2at the top, so:
(0 ≤ y ≤ x2).
The new set of inequalities, given that we are reversing the order of integration, is:
(0 ≤ x ≤ 1)
(0 ≤ y ≤ x2).
Step 3: Write the new integral with the inequalities from Step 2. Don’t forget to reverse “dx” and “dy”.
Step 4: Integrate as usual. For this double integral, you’ll need to integrate twice: once with respect to y, then with respect to x.
Ikenaga, B. Interchanging the Order of Integration. Retrieved July 3, 2020 from: http://sites.millersville.edu/bikenaga/calculus/interchanging-the-order/interchanging-the-order.html
Ram, B. Engineering Mathematics. Pearson, 2009.
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