# integral application

• Nov 15th 2006, 09:08 PM
synnexster
integral application
Im really not sure how to even approach this problem... it says

Suppose the outdoor temperature on a particular day is given by T(t) = 60 + 16*(sin((pi*t)/24))^2 where time is measured in hours and t =0 corresponds to midnight. What is the average temperature for a 24-hour period?
• Nov 16th 2006, 04:14 AM
topsquark
Quote:

Originally Posted by synnexster
Im really not sure how to even approach this problem... it says

Suppose the outdoor temperature on a particular day is given by T(t) = 60 + 16*(sin((pi*t)/24))^2 where time is measured in hours and t =0 corresponds to midnight. What is the average temperature for a 24-hour period?

There are two ways to approach this depending on what class you are in.

The first is to simply note that the average of $sin^2(\theta)$ over a full period is 1/2.

The integral formula is:
$\bar{f(x)} = \frac{1}{X} \int_0^Tdxf(x)$
where X is the interval over which you are integrating.

So
$\bar{T} = \frac{1}{24} \int_0^{24}dt \left ( 60 + 16*sin \left ( \frac{\pi t}{24} \right )^2 \right )$

I'll leave it to you to do the integration. If you need help, feel free to let me know.

-Dan