Video: Evaluating a Definite Integral Using the Property of Addition of Two Definite Integrals over Two Adjacent Intervals

If ∫_(βˆ’5)^(2) 𝑓(π‘₯) dπ‘₯ = βˆ’2.4 and ∫_(βˆ’5)^(βˆ’1) 𝑓(π‘₯) dπ‘₯ = βˆ’1.4, find ∫_(βˆ’1)^(2) 𝑓(π‘₯) dπ‘₯.

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Video Transcript

If the integral of 𝑓 of π‘₯ with respect to π‘₯ between negative five and two is negative 2.4 and the integral of 𝑓 of π‘₯ with respect to π‘₯ between negative five and negative one is negative 1.4, find the integral of 𝑓 of π‘₯ with respect to π‘₯ between negative one and two.

In order to solve this problem, we need to recall the following rule. The integral of any function between two limits π‘Ž and 𝑐 is equal to the integral of the same function between the limits π‘Ž and 𝑏 plus the integral of the function between the limits 𝑏 and 𝑐. In our question, we have three different integers for our limits: two, negative five, and negative one. The smallest of these is negative five. So we will let π‘Ž equal negative five. The middle value is equal to negative one, so 𝑏 is equal to negative one. The largest value is two, so we will let 𝑐 equal two.

This means that the integral of 𝑓 of π‘₯ between negative five and two is equal to the integral of the same function between negative five and negative one plus the integral between negative one and two. We’re told in the question that the value of the left-hand side is equal to negative 2.4. The value of the first term on the right-hand side is equal to negative 1.4. We’re trying to calculate the integral of 𝑓 of π‘₯ with respect to π‘₯ between negative one and two.

We can solve this equation by adding 1.4 to both sides of the equation. Negative 2.4 plus 1.4 is equal to negative one. And negative 1.4 plus 1.4 equals zero. We can therefore conclude that the integral of 𝑓 of π‘₯ with respect to π‘₯ between negative one and two is equal to negative one.

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