31. Section 5.5 Change as the
Integral of a Rate
Essential Question
• How do you find total distance traveled given
the velocity function?
Integrals of Rates
• If you are given a rate and asked to find the
integral, you are finding the total change
• i.e. If you are given people/hr and take the
integral, you end up with total people.
• Or if you are given water/min and take the
integral, you end up with total amount of
water
Velocity
• If you are given a position function, how do you
find velocity?
• Differentiate
• So what if we a given a velocity function and
want position, what would we do?
• Integrate!!
v(t )dt

• Net Distance =
• Total Distance =  v(t ) dt
• (because even if we are going left, we want to
add in distance)
t2
t1 t2
t1
Example
• The velocity of a particle is v(t )  t 3  10t 2  24t
• Find displacement over [0,4][4,6] and [0,6]
and total distance over [0,6]
1 4 10 3
3
2
2
(
t

10
t

24
t
)
dt

t

t

12
t
C

4
3
4
1
10
[0, 4]  t 4  t 3  12t 2  42.667
4
3
0
6
1
10
[4, 6]  t 4  t 3  12t 2  6.667
4
3
4
6
1
10
[0,6]  t 4  t 3  12t 2  36
4
3
0
TD[0,6]  42.667  6.667  49.333
Example – Marginal Cost
• Marginal cost is the derivative (rate) of the cost function.
• The marginal cost of producing x computer chips is
C '( x)  150 x 2  3000 x  17500 in dollars per thousand chips.
Find the cost of increasing production from 10000 chips to
15000 chips. Then find the total cost of producing 15000 chips
if startup cost is $35000.
15
a.
2
(150
x
 3000 x  17500)dx

10
50 x  1500 x  17500 x
3
2
15
10
 $18,750
15
b.
2
(150
x
 3000 x  17500)dx  35000

0
50 x  1500 x  17500 x
3
2
15
0
 $93,750  $35,000  $128,750
Assignment
• Pg. 341: #1-23 odd