How much money did the soda machines generate during the second semester?

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SUMMARY

The discussion centers on calculating the total income generated by soda machines installed in a West Texas school cafeteria, using the income function f(t) = 500t / (5t² + 2). The correct limits of integration for the second semester, which runs from January 1 to May 31, are established as 5 months (from t=1 to t=6). The initial misunderstanding involved using incorrect limits, specifically t=6 as the upper limit, which does not account for the revenue generated in the first month of operation.

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Homework Statement


In a West Texas school district the school
year began on August 1 and lasted until May
31. On August 1 a Soft Drink company in-
stalled soda machines in the school cafeteria.
It found that after t months the machines
generated income at a rate of
f(t) =
500t/ (5t^2 + 2)
dollars per month. Find the total income,
$Tsec, produced during the second semester
beginning on January 1.


Homework Equations



I know to find the income we first have to find the integral for the function t. I did this and got 50 ln [5x^2 +2].
I set u = 5t^2 +2, so du = 10t dt

The Attempt at a Solution



I used 6 as my upper limit and 1 as my lower limit but I can't get the right answer. Can someone please tell me what is the right limit of integration to use in this case?
Thank you so much for your help!
 
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Your limits are a little off. Why did you use 6? The machine was functioning for 5 months from Aug 1 to Jan 1 the next year. And it's not supposed to be 0 either for the lower limits. That ignores the revenue for the first month.
 

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