Video: Evaluating a Definite Integral Using the Property of Addition of the Integral of Two Functions over the Same Interval

The function 𝑓 is continuous on [βˆ’4, 4] and satisfies ∫_(0) ^(4) 𝑓(π‘₯) dπ‘₯ = 9. Determine ∫_(0) ^(4) [𝑓(π‘₯) βˆ’ 6] dπ‘₯.

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Video Transcript

The function 𝑓 is continuous on the closed interval negative four to four and satisfies the definite integral between zero and four of 𝑓 of π‘₯ with respect to π‘₯ is equal to nine. Determine the definite integral between zero and four of 𝑓 of π‘₯ minus six with respect to π‘₯.

To answer this question, we’ll begin by recalling some basic properties of definite integrals. Firstly, we know that the integral of the sum of two functions is equal to the sum of the integral of each respective function. Similarly, the integral of the difference of two functions is equal to the difference of the integrals of those functions. This means we can begin by rewriting our integral as the difference between the integral between zero and four of 𝑓 of π‘₯ with respect to π‘₯ and the integral between zero and four of six with respect to π‘₯. We also recall that for constant function 𝑐, the definite integral between π‘Ž and 𝑏 of 𝑐 with respect to π‘₯ is equal to 𝑐 times 𝑏 minus π‘Ž.

Now, our 𝑐 is equal to six, and π‘Ž is zero, 𝑏 is four. So the definite integral between zero and four of six with respect to π‘₯ is simply six times four minus zero, which is equal to 24. As per the question, we now replace the integral between zero and four of 𝑓 of π‘₯ with respect π‘₯ with nine. And we find that the integral of 𝑓 of π‘₯ minus six between the limits of zero and four with respect to π‘₯ is nine minus 24. That’s equal to negative 15. So, despite not knowing the function 𝑓, we see that the definite integral between zero and four of 𝑓 of π‘₯ minus six with respect to π‘₯ is negative 15.

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