Section 5.3 The Fundamental Theorem and Interpretations

The symbol is used to remind us that an integral is a sum of the terms which are f(x) times a small difference in x, or dx dx is part of the integral symbol meaning “with respect to x” The unit measurement for an integral is essentially the product of the units for f(x) and the units for x Let’s interpret this in terms of our example from 5.1 concerning velocity and time

Example Suppose C(t) represents the daily cost of heating your house, measured in dollars per day, where t is measured in days since 12 am January 1, Interpret Give units.

Suppose that the rate that people are entering the mall on the day after thanksgiving is give by r(t) where t is measured in hours and t = 0 is when the mall opens at 6am. Write a definite integral which represents the number of people for the following time intervals: From open until close (8pm) During lunch, approximately 11:30 until 2pm

If we have a velocity function, v = f(t), the change in position is given by the definite integral Now if we define F(t) to be the position function, then the change in position can also be written as F(b) – F(a) Finally, we know that the position F and velocity f are related using derivatives: F’(t) = f(t) Based on this relationship we get the following theorem

If f is continuous on the interval [a,b], and f(t) = F’(t), then or we have the following interpretation Let’s verify this with a problem from last time The Fundamental Theorem of Calculus Total Change in F(t) between t = a to t = b

How do you compute an average of n numbers? Now suppose you have a continuously varying function, for example temperature as a function of time (in hours) If we wanted to know the average temperature for the day and divide it by the number of hours in a day Now to get more accurate we would add up more and more times/temperatures Thus we would have a Riemann sum We could then turn this into a definite integral and then divide by the time period to find the average This will work for any continuous function

Average Value of a Function If f is continuous on [a, b] then the average value of f from a to b is given by Find the average value of from 0 to 2.