Calculus How To

Domain of Integration

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When you integrate over a closed interval [a, b] of some function, the interval is called the domain of integration. For example:

The domain of integration for this function is the closed interval [1, 3].

A domain of integration can be infinite (i.e. from -∞ to ∞), as the following improper integral shows:

An example of an improper integral, with an infinite domain of integration.

Improper integrals can’t be calculated directly; They are calculated as limits of ordinary integrals.

Things get a little more complicate in three dimensions, but most of the time the area on the base of the object is the domain of integration.


The domain of integration on a 3D shape is usually the base (shown by red lines in this image).

How to Sketch a Domain of Integration

Drawing a domain for any integral is easy if you only have one integral (see Step 1 below for an example). Things get a little more complicated for multiple integrals, but if you break it down into steps it becomes a lot less challenging.

Example problem: Sketch the domain of integration for the following iterated integral:
iterated integral 2

Step 1: Draw the bounds of integration for the first integral. The bounds are given as x = 0 to 1, so:
step 1 sketch domain of integration

Step 2: Draw the bounds of the second integral on the same graph from Step 1. Note: If the bounds of integration aren’t integers (the second integral here has e, Euler’s number), you may want to use a graphing calculator (I used the one at to draw this graph) so you can more easily sketch the shape.

Step 3: Find the shaded area that meets the definition of both integrals. For this example, you’re only shading the area from 0 to 1 that is also within [e, ex].

That’s it!


Liu, G. Calculus of Several Variables. Retrieved August 31, 2020 from:
Rogawski, J. (2007). Multivariable Calculus. W. H. Freeman.
Zhou, A. Problem. Retrieved August 31, 2020 from:

Stephanie Glen. "Domain of Integration" From Calculus for the rest of us!

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