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Calculus/Integral Calculus

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Question
The promoters of a county fair estimate that t hours after the gates open at 9:00 AM, visitors will be entering the fair at the rate of Ní(t) people per hour. Find an expression for the number of people who will enter the fair between 11:00 AM and 1:00 PM.

Answer
If the rate at which they will enter is a derivative of the number there, then the function is an exponential equation.  The equation would be N(t) = e^t where t is in hours.

Now if it opens at 9am, then the times looked at are t=2 and t=4 since 11am is two hours later and 1pm is 4 hours later.

Since the integral of e^t is e^t, the answer would be e^4 - e^2.
Now e^2 = 7.389056099 and e^4 = 54.59815003, so the difference is 47.20909393.
Since this is the number of people, it would be rounded to the nearest integer, so n = 47.  

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