Section 5'3 The Fundamental Theorem and Interpretations

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Section 5.3The Fundamental Theorem and
Interpretations
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• 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
• Lets interpret this in terms of our example from
  5.1 concerning velocity and time

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• 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

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The Fundamental Theorem of Calculus

• If f is continuous on the interval a,b, and
  f(t) F(t), then
• or we have the following interpretation
• Lets verify this with a problem from last time

Total Change in F(t)between t a to t b
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• Lets look at 3 and 6 from the book (pg 261)
• 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 1130 until 2pm

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• 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

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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