Calculus How To

Iterated Integrals

Integrals >

What is an Iterative Process?
Iterated Integral

What is an Iterative Process?

An iterative process is a process which is run over and over again repeatedly, to ultimately reach or approach a desired result. Each cycle, or repetition, is called an iteration. An iterative process may include a finite (fixed) number of iterations with a definite stopping point, or it may go on infinitely.

Iterative Process Examples

As a very simple example, counting by twos is an iterative process, because you’re adding 2 each time, over and over again.

The Koch snowflake is a more complicated example of an iterative process. The first iteration is an equilateral triangle; each successive iteration is formed by adding smaller equilateral triangles to the first.

A recursive formula is a sequence or quantity that is described by a iterative process.

One recursive formula is the one which describes the Fibonacci sequence. The Fibonacci sequence can be defined as:

  • F0 = 0
  • F1 = 1
  • Fn = Fn-1 + Fn-2

To find the value of a given Fibonacci number, you can run an iterative process, starting from F0 and F1 and finding the next number by repeatedly using the formula Fn = Fn-1 + Fn-2.

Using this formula, F2 is 0 + 1 = 1. To find F4, though, you would need to run several iterations of the forumla—first find F2 = 1 and F3 = F2 + F1 = 1 + 1 = 2. For any Fibonacci number n >2, you would have to run n – 1 calculations to find out its value using only the definition above.

Calculating compound interest is another example of iterative processes. If interest is compounded yearly, we find the total interest accumulated in ten years by multiplying, year by year, the interest + 100% by the previous years balance.

Iterated Integrals

An iterated integral has the general form (Rogawski, 2007):
iterated integral

The expression is made up of an “inner integral” and one or more outer integrals. An iterated integral with two integrals is called a double integral; A triple integral is a three integral expression.

Solving the Iterated Integral

An iterated integral is worked much in the same way that inner functions and outer functions are worked in the chain rule for derivatives: you start by evaluating the inner function (or in this case, the inner integral), then work your way out. In other words, you’re performing iterative integration. In the generic example given above, you would integrate with respect to y first (using c, d as the bounds of integration), then work the new integral with respect to x (using a, b as the bounds of integration).

For an example, see: Solving a Double Integral or Solving a Triple Integral.

One of the surprising benefits of iterated integrals is that you can change the order of integration if, for example, the inner integral is impossible to evaluate. While “regular” integration is like slicing a loaf of bread along it’s length, changing the order of integration allows you to slice across its width instead.

Fubini Theorem’s

Theorems which relate multiple integrals (integrated over subsets of ℝ) to iterated integrals are normally called Fubini Theorems (Swarz, 2001). In simple terms, Fubini’s theorem states:

“…when we have a ‘nice’ function, that the n-dimensional multiple integral of this function is the same as the n-fold iterated integral” (Tollas, 2007).


Iteration in Matlab. Retrieved from on January 8, 2018.
Rogawski, J. Multivariable Calculus. W. H. Freeman. 2007.
Rudin, W., Real and complex analysis, 1970. McGraw-Hill Education.
Swarz, C. Introduction to Gauge Integrals. World Scientific. 2001.
Section 12.2/12.3: Iterated Integrals Double Integrals over General Regions. Retrieved July 3, 2020 from:
Tollas, L. (2007). Iterated Integrals. Article posted on Reed College website. Retrieved July 3, 2020 from:

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published. Required fields are marked *