# 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
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!
– 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.


 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.