Average Integral Problem?
In a certain city the temperature (in degrees Fahrenheit) t hours after 9am was approximated by the function: Integral: 40+11Sin(Pi(x)/12)
Find the average temperature during the period from 9 am to 9 pm.
In a certain city the temperature (in degrees Fahrenheit) t hours after 9am was approximated by the function: Integral: 40+11Sin(Pi(x)/12)
Find the average temperature during the period from 9 am to 9 pm.
Comments
So, we want the average value of f(t) = 40 + 11 sin(πt/12) for t in [0, 12].
This equals
[1/(12 - 0)] * integral(0 to 12) f(t) dt
= (1/12) * integral(0 to 12) [40 + 11 sin(πt/12)] dt
= (1/12) * [40t - 11 (12/π) * cos(πt/12)] {for t = 0 to 12}
= [40t/12 - (11/π) * cos(πt/12)] {for t = 0 to 12}
= [40 + 11/π] - [0 - 11/π]
= (40 + 22/π) degrees Fahrenheit.
I hope this helps!