Give me an example of an application problem that uses the mean value theorem of integrals as opposed to the mean value theorem for derivatives. I'm having trouble finding a difference.

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Give me an example of an application problem that uses the mean value theorem of integrals as opposed to the mean value theorem for derivatives. I'm having trouble finding a difference.

Mathematics
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The biggest difference is derivatives deal with slope, dy/dx and integrals deal with areas under a curve
first understand the difference between derivatives and integrals: they are inverse operations like multiplying and dividing. Integral question. Find the average value of given function on prescribed interval x^2 - x + 1 on [-1,2]
So here's and example: On a certain day the temperature t hours past midnight was T(t)=60+5sin((pi/6)(t-10)). What was the average temperature between noon and mignight of the same day?

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Sorry you wanted application: f(x) = sq rt (9 - x^2) on [-3,3] Evaluate integral as part of circle.
At first, I would think its just mean value theorem for derivatives, but the answer is integrals.
Actually I mean a story problem.
Mean Value Theorem for derivatives: car travels average velocity 60 mi/h; at some point the speedometer had read 60 m/h. Integrals is similar and gives a geometric interpretation of the theorem and makes it easier to understand. For example on an elementary question with multiplication and division the question or problem doesn't change for each, you just decide whether multiplication is appropriate or division is appropriate.

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