1. Do Now - #22 and 24 on p.275
Graph the function over the interval. Then (a) integrate the
function over the interval and (b) find the area of the region
between the graph and the x-axis.
Possible graph window: [0, 2] by [–5, 3]
An antiderivative of the function:
(a)

2. Do Now - #22 and 24 on p.275
Graph the function over the interval. Then (a) integrate the
function over the interval and (b) find the area of the region
between the graph and the x-axis.
Possible graph window: [0, 2] by [–5, 3]
(b) Area =

3. Do Now - #22 and 24 on p.275
Graph the function over the interval. Then (a) integrate the
function over the interval and (b) find the area of the region
between the graph and the x-axis.
Possible graph window: [0, 3] by [–5, 5]
An antiderivative of the function:
(a)

4. Do Now - #22 and 24 on p.275
Graph the function over the interval. Then (a) integrate the
function over the interval and (b) find the area of the region
between the graph and the x-axis.
Possible graph window: [0, 3] by [–5, 5]
(b) Area =

5. Average Value of Functions
Section 5.3b

6. First, remind me… How do you find the average of n numbers?
 Sum the numbers and divide by n !!!
Our New Challenge:
How do we find the average value of an
arbitrary function f over a closed interval [a, b]???

7. Does this look familiar???
We take a large sample of n numbers from regular subintervals
of the interval [a, b], taking number from each subinterval
of length
The average value of the n sampled values is
Does this look familiar???

8. So what happens as n approached infinity?
We take a large sample of n numbers from regular subintervals
of the interval [a, b], taking number from each subinterval
of length
The average value of the n sampled values is
Can you say RIEMANN SUM???
So what happens as n approached infinity?

9. Definition: Average (Mean) Value
If is integrable on [a, b], its average (mean) value
on [a, b] is

10. The Mean Value Theorem for Definite Integrals
If f is continuous on [a, b], then at some point c in [a, b],
The value f (c) in the MVT is, in a
sense, the average (or mean)
height of f on [a, b]. When f > 0,
the area of the shaded rectangle a c b
is the area under the graph
of f from a to b.
b – a

11. The average value of the
Find the average value of the given function on [0, 3]. Does the
function actually take on this value at some point on the interval?
Use NINT!!!
The average value of the
function on [0, 3] is 1.
The function assumes this value when
Since this x value lies in the interval [0, 3], the function does
assume its average value in the given interval.

12. The rectangle with base [0, 3] and with height equal to 1 (the average
Find the average value of the given function on [0, 3]. Does the
function actually take on this value at some point on the interval?
The average value of the function on [0, 3] is 1.
The function assumes this value when 4
The rectangle with base [0, 3] and
with height equal to 1 (the average
value of the function) has area
equal to the net area between f
and the x-axis from 0 to 3. 4 –5

13. Quality Practice Problems
Find the average value of the given function on the given interval.
At what point(s) in the interval does the function assume its
average value?
Find the point:
In the given interval:

14. Quality Practice Problems
Find the average value of the given function on the given interval.
At what point(s) in the interval does the function assume its
average value?
Find the point:
Both are in the given interval!