Calculus How To

Additive Interval Property

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The additive interval property (sometimes called the additive integral property) tells us that we can add up parts of an integral to get a whole.

Formally, the rule is stated as [1]:
additive interval rule definition

All this is saying is that a definite integral (i.e. an area under a curve) from point a to point b can be split at a point “c“. The two pieces or “step integrals” can be added together to get the total area. More specifically, a definite integral evaluated over the closed interval [a, b] is the same as the sum of parts of the integrals (split at point c) over [a, c] and [c, b] when a ≤ c ≤ b.

The integral can be split at any point, and you can have more than two pieces.

Example of the Additive Interval Property

additive interval property example 1

Integral A = Integral B + Integral C.

The above image shows the definite integral of f(x) = 2x on the interval [0, 3]. Calculating this definite integral using the integral power rule, we get:
integral of 2x

We can split the integral at x = 2 and get the same result:
integral of 2x. 2

Using the Additive Property for Integrals

The addition problem can be useful to work backwards, finding intervals that have a certain area. For example, if you knew that the whole integral had an area of 9, and one part of the integral had an area of 3, you could conclude that the “missing” interval had an area of 6. From there, you could work out the missing bounds of integration.

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[1] Larson, R. & Edwards, B. (2017). Calculus, 11th Edition. Cengage Learning.

Stephanie Glen. "Additive Interval Property" From Calculus for the rest of us!

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