Question Video: Evaluating a Definite Integral Using the Property of Addition of Two Definite Integrals over Two Adjacent Intervals Mathematics • Higher Education

If ∫_(βˆ’2)^(6) 𝑓(π‘₯) dπ‘₯ = 1 and ∫_(βˆ’2)^(9) 𝑓(π‘₯) dπ‘₯ = 11, find ∫_(6)^(9) 𝑓(π‘₯) dπ‘₯.

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Video Transcript

If the integral of 𝑓 of π‘₯ with respect to π‘₯ between negative two and six is equal to one and the integral of 𝑓 of π‘₯ with respect to π‘₯ between negative two and nine is equal to 11, find the integral of 𝑓 of π‘₯ with respect to π‘₯ between six and nine.

In order to answer this question, we need to recall that the integral of 𝑓 of π‘₯ between π‘Ž and 𝑐 is equal to the integral of the same function between π‘Ž and 𝑏 plus the integral of the function between 𝑏 and 𝑐. In our question, we have three different limits, negative two, six, and nine. As negative two is the smallest of these three values, we will let π‘Ž equal negative two. The middle value is six, so we will let 𝑏 equal six. Finally, the largest value is nine, so 𝑐 is equal to nine.

The integral of 𝑓 of π‘₯ between negative two and nine is equal to the integral of 𝑓 of π‘₯ between negative two and six plus the integral of 𝑓 of π‘₯ between six and nine. We know that the term on the left-hand side is equal to 11. The first term on the right-hand side is equal to one. The second term on the right-hand side, the integral of 𝑓 of π‘₯ between six and nine, is what we’re trying to calculate. In order to solve this equation, we can subtract one from both sides of the equation. 11 minus one is equal to 10. We can therefore conclude that the integral of 𝑓 of π‘₯ with respect to π‘₯ between six and nine is equal to 10.

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