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# Integration question Watch

1. Hey,
Looking at this integrtion question

The temperature (in degrees cenigrade) measured at a weather station on 21st Sept is given by

T(t) = 15 - 10cos^2(pi*t/24) - t/10

0 < t < 24

where t is the time measured in hours. Find average temperature over the entire day, and average during the hours of daylight 6<t<18

It's just integrating the expression I can't do, the worst part is I have my working from when I have done it before but I haven't copied it all down and now I can't see how I integrated that expression last time to get the answer :|

Thanks for any help
2. (Original post by RHCPfan)
Hey,
Looking at this integrtion question

The temperature (in degrees cenigrade) measured at a weather station on 21st Sept is given by

T(t) = 15 - 10cos^2(pi*t/24) - t/10

0 < t < 24

where t is the time measured in hours. Find average temperature over the entire day, and average during the hours of daylight 6<t<18

It's just integrating the expression I can't do, the worst part is I have my working from when I have done it before but I haven't copied it all down and now I can't see how I integrated that expression last time to get the answer :|

Thanks for any help
Use double angle formulae for

and rearrangle for cos squared.

and
3. (Original post by RDKGames)
Use double angle formulae for

and rearrangle for cos squared.

and
Ah yeah so that gives me

T(t) = 15 - 5 - 5cos(pi*t/12) - t/10
= 10 - 5cos(pi*t/12) - t/10

thanks!

I'm just a little unsure how to do the, - 5cos(pi*t/12)? Sorry my brain is failing as i'm sitting looking at my previous correct answer for this!
4. (Original post by RHCPfan)
Ah yeah so that gives me

T(t) = 15 - 5 + 5cos(pi*t/12) - t/10
= 10 + + 5cos(pi*t/12) - t/10

thanks!

I'm just a little unsure how to do the, + 5cos(pi*t/12)? Sorry my brain is failing as i'm sitting looking at my previous correct answer for this!
Reverse chain rule. Cos goes to sine and you divide the 5 by the coefficient of t; in this case pi/12
5. (Original post by RDKGames)
Reverse chain rule. Cos goes to sine and you divide the 5 by the coefficient of t; in this case pi/12
Ah yeah! Thanks so now I have it

(-60/pi)sin(pi*t/12)

Thank you
6. (Original post by RDKGames)
Reverse chain rule. Cos goes to sine and you divide the 5 by the coefficient of t; in this case pi/12
The chain rule in general doesn't work in reverse: the method that you have described only works when the derivative of the inner function is a constant, or in other words, the inner function is linear. What you are really doing is making a substitution and skipping out the steps that you can do in your head.
7. (Original post by HapaxOromenon3)
The chain rule in general doesn't work in reverse: the method that you have described only works when the derivative of the inner function is a constant, or in other words, the inner function is linear. What you are really doing is making a substitution and skipping out the steps that you can do in your head.
Exactly. No need to fuss around explaining this to him if he had done it before.

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