Integral absolute minimum/maximum

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Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must the integral G lie?

I thought it was m(3) for abs min.

and M(3) for max. Is it 2 instead since 1-->3 is 2? Not sure. thanks for any help!

Re: Integral absolute minimum/maximum

What is a (Riemann) integral?

Let f be a function on [a,b] and be a partition P of [a,b]. Let where is the minimum (actually greatest lower bound) of f on . Similarly, where is the maximum of f on .

The integral is that unique number with for every partition P.

That's really the definition of integral. The great fundamental theorem of calculus allows "easy" computation of G.

In particular a partition with n=1, has and and so

So for your specific problem

Re: Integral absolute minimum/maximum

All that is correct, but you should probably just remember the general rule that if a function has a bound, , then it's integral (if it exists) is bounded by M times the length of the interval, . If you draw a graph and compare the two areas, it should be clear.

- Hollywood

Re: Integral absolute minimum/maximum

Thank you both! I had a feeling you had to subtract 3 and 1 but I wasn't 100%. Both of your explanations really make it clear though!