Using the Definite Integral _21804_

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					Clicker Question 1

   What is an antiderivative of f(x) = (5x – 3) ?
    – A. (5/( + 1))(5x – 3) +1
    – B. (1/( + 1))(5x – 3) +1
    – C. 5(5x – 3) - 1
    – D. (1/(5( + 1)))(5x – 3) +1
    – E. (5( + 1))(5x – 3) +1
Clicker Question 2

   What is an antiderivative of g(x) = x / (x2 + 1)?
    – A. x ln(x2 + 1)
    – B. (1/2) ln(x2 + 1)
    – C. 1 / (x2 + 1)2
    – D. (-1/4) / (x2 + 1)2
    – E. 2 ln(x2 + 1)
Concerning Definite Integrals
and Substitution (1/28/11)

   If you use substitution and the Fundamental
    Theorem to evaluate a definite integral, there
    are two possible approaches:
    –   Go back to the original variable and evaluate at
        the endpoints as usual, or
    –   Never return to the original variable! Instead,
        change the endpoints to correspond to your new
        variable, and then stay with that variable.
Using the Definite Integral

   This semester we shall study numerous
    applications of the definite integral to
    geometry, physics, economics, probability,
    and so on.
   Remember that whenever you want to
    “add up” the values of a function over some
    interval, the definite integral may well be the
    ticket!
   We start with an easy application:
     – Average value of a function on an interval
Average Value of a Function
on an Interval

   To find the average value of a list of numbers, you
    add them up and divide by how much is there.
   It’s the exact same for functions: add up the values
    of the function on the interval in question and then
    divide by how much is there (i.e., the length of the
    interval).
   Thus the average value of f on [a, b] is
                b

                 f ( x)dx
                a
                             ba
Example of Average Value

   What is the average value of sin(t) on the interval
    [0, ] ? Look at the picture and make a guess.
   The answer is
        

         sin(t )dt
        0
                             = 2 /   .637
                       0
   Check that this answer makes sense. (The average
    value on a graph is the average height, i.e. the height
    whose rectangle has the same area as the area under
    the curve.)
Clicker Question 3

   What is the average value of f(x) = x2 on the
    interval [0, 4]?
     – A. 8
     – B. 21 1/3
     – C. 5 1/3
     – D. 6 2/3
     – E. 7 2/3
Assignment for Monday

   On page 407, do Exercises 53, 55, 59, and
    61.
   Read Section 6.5.
   On page 445, do Exercises 1-11 odd and 17.

				
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