# Question Video: Subtracting Two Numbers up to 9 Mathematics • Higher Education

Suppose that on [−2, 5], the values of 𝑓 lie in the interval [𝑚, 𝑀]. Between which bounds does ∫_(−2) ^(5) 𝑓(𝑥) d𝑥 lie?

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### Video Transcript

Suppose that on the closed interval negative two to five, the values of 𝑓 lie in the closed interval lowercase 𝑚 to capital 𝑀. Between which bounds does the definite integral between negative two and five of 𝑓 of 𝑥 with respect to 𝑥 lie?

We recall one of the comparison properties of integrals. It tells us that if 𝑓 of 𝑥 is greater than or equal to lowercase 𝑚 and less than or equal to capital 𝑀 for values of 𝑥 greater than or equal to 𝑎 and less than or equal to 𝑏. Then 𝑀 times 𝑏 minus 𝑎 is less than or equal to the definite integral between 𝑎 and 𝑏 of 𝑓 of 𝑥 with respect to 𝑥, which in turn is less than or equal to capital 𝑀 times 𝑏 minus 𝑎. In other words, given that lower case 𝑚 is the absolute minimum of 𝑓 and uppercase 𝑀 is the absolute maximum of 𝑓, the area under the graph of 𝑓 is greater than the area of the rectangle with a height lower case 𝑚. But it’s less than the area of the rectangle with a height uppercase 𝑀.

In this example, we’re going to let 𝑎 be equal to negative two and 𝑏 be equal to five. Then, we see that 𝑀 times five minus negative two is less than or equal to the definite integral between negative two and five of 𝑓 of 𝑥 with respect to 𝑥 which, in turn, is less than or equal to capital 𝑀 times five minus negative two. Five minus negative two is seven. And so, we see that our definite integral must be greater than or equal to seven 𝑚 and less than or equal to seven uppercase 𝑀.