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? I know that you would use the mean value theorem for integrals to solve this which I know how, but why?

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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? I know that you would use the mean value theorem for integrals to solve this which I know how, but why?

Mathematics
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Wouldn't you just use the mean value thoerem for derivatives since you're trying to find the average temperature? Not time*temperature as the area under the curve of this function suggests?
you're right we are looking for average temperature and we are looking at area under curve but not of T but T' _>\[=1/12\int\limits_{0}^{12}T 'dt\] which really is (T(12)-T(0))/12
think of it as Temp/time * time

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