Total Change and Approximation
Objectives:
Assignment:
1. To use integrals of rates
of change to find total
change
• HW Worksheet: 1-10
2. To numerically
approximate definite
integrals
• HW Worksheet: 11-14
Warm Up
The rate at which rainwater flows into a drainpipe is
modeled by the function 𝑅, where 𝑅 𝑡 = 20 sin
𝑡2
35
cubic feet per hour, 𝑡 is measured in hours, and
0 ≤ 𝑡 ≤ 8. The pipe is partially blocked, allowing
water to drain out the other end of the pipe at a rate
modeled by 𝐷 𝑡 = −0.04𝑡 3 + 0.4𝑡 2 + 0.96𝑡 cubic
feet per hour, for 0 ≤ 𝑡 ≤ 8. There are 30 cubic
feet of water in the pipe at 𝑡 = 0. How many cubic
feet of water flow into the pipe during the 8-hour
time interval 0 ≤ 𝑡 ≤ 8?
Objective 1
You will be able to
use integrals of rates
of change to find
total change
Total Change
The Fundamental Theorem of Calculus says
𝑏
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎), where 𝐹 is any
𝑎
antiderivative of 𝑓.
In other words 𝐹′(𝑥) = 𝑓(𝑥), so…
𝑏
𝑏
𝐹′(𝑥) 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎)
𝑓(𝑥) 𝑑𝑥 =
𝑎
𝑎
Rate of Change
Total Change
Total Change
The integral of a rate of change of a quantity
𝐹′ 𝑥 gives the total change or net
change in that quantity on the interval
𝑎, 𝑏 .
𝑏
𝑏
𝐹′(𝑥) 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎)
𝑓(𝑥) 𝑑𝑥 =
𝑎
𝑎
Rate of Change
Total Change
Total Change
The integral of a rate of change of a quantity
𝐹′ 𝑥 gives the total change or net
change in that quantity on the interval
𝑎, 𝑏 .
Final Amount
𝑏
𝑏
𝐹′(𝑥) 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎)
𝑓(𝑥) 𝑑𝑥 =
𝑎
𝑎
Rate of Change
Initial Amount
Total Change
If 𝑉(𝑡) is the volume of water in a reservoir at
time 𝑡, then 𝑉′(𝑡) is the rate at which the
volume flows into the reservoir at time 𝑡.
𝑡2
𝑡1
𝑉 ′ (𝑡) 𝑑𝑡 = 𝑉 𝑡2 − 𝑉 𝑡1
The total change in
the amount of water
in the reservoir
from 𝑡1 to 𝑡2 .
Exercise 1
The rate at which rainwater flows into a drainpipe is
modeled by the function 𝑅, where 𝑅 𝑡 = 20 sin
𝑡2
35
cubic feet per hour, 𝑡 is measured in hours, and
0 ≤ 𝑡 ≤ 8. The pipe is partially blocked, allowing
water to drain out the other end of the pipe at a rate
modeled by 𝐷 𝑡 = −0.04𝑡 3 + 0.4𝑡 2 + 0.96𝑡 cubic
feet per hour, for 0 ≤ 𝑡 ≤ 8. There are 30 cubic
feet of water in the pipe at 𝑡 = 0. How many cubic
feet of water flow into the pipe during the 8-hour
time interval 0 ≤ 𝑡 ≤ 8?
Exercise 2
The pipe from the previous Exercise can hold 50
cubic feet of water before overflowing. For 𝑡 > 8,
water continues to flow into and out of the pipe at
the given rates until the pipe begins to overflow.
Write, but do not solve, an equation involving one
or more integrals that gives the time 𝑤 when the
pipe will begin to overflow.
Rate of flow into pipe:
Rate of flow out of pipe:
𝑡2
𝑅 𝑡 = 20 sin
35
𝐷 𝑡 = −0.04𝑡 3 + 0.4𝑡 2 + 0.96𝑡
O
Positive Direction
Negative Direction
Rectilinear Motion
Rectilinear motion is
movement along a straight
line with respect to an origin
point.
Positive Direction
O
Negative Direction
𝒔 𝒕
Rectilinear Motion
𝒗 𝒕
𝒂 𝒕
𝒔 𝒕 = Position
𝒔′ 𝒕 = 𝒗 𝒕 = Velocity
𝒔′′ 𝒕 = 𝒗′ 𝒕 = 𝒂 𝒕 = Acceleration
𝒔 𝒕
Rectilinear Motion
𝒗 𝒕
𝒂 𝒕
𝒂(𝒕) 𝒅𝒕 =
𝒗(𝒕) 𝒅𝒕 = 𝒔 𝒕 = Position
𝒂(𝒕) 𝒅𝒕 = 𝒗 𝒕 = Velocity
𝒂 𝒕 = Acceleration
Displacement
If an object moves along a straight line
according the position function 𝑠(𝑡) at time 𝑡,
then its velocity is 𝑣 𝑡 = 𝑠 ′ (𝑡).
𝑡2
𝑣(𝑡) 𝑑𝑡 = 𝑠 𝑡2 − 𝑠 𝑡1
𝑡1
The change in the
position, called
displacement, of
the object from
𝑡1 to 𝑡2 .
Displacement
If an object moves along a straight line
according the position function 𝑠(𝑡) at time 𝑡,
then its velocity is 𝑣 𝑡 = 𝑠 ′ (𝑡).
Final Position
𝑡2
𝑡2
𝑣(𝑡) 𝑑𝑡 = 𝑠 𝑡2 − 𝑠 𝑡1
𝑡1
𝑠 𝑡2 = 𝑠 𝑡1 +
𝑣(𝑡) 𝑑𝑡
𝑡1
Initial Position
Exercise 3
The figure at the left shows
the graph of 𝑓′, the derivative
of a twice-differentiable
function 𝑓, on the interval
−3,4 . The graph of 𝑓′ has
horizontal tangents at
𝑥 = −1, 𝑥 = 1, and 𝑥 = 3.
The areas of the regions
bounded by the 𝑥-axis and
the graph of 𝑓′ on the
intervals −2,1 and 1,4 are
9 and 12, respectively.
Given that
𝑓 1 = 3, write an
expression for
𝑓(𝑥) that involves
an integral. Find
𝑓(4) and 𝑓(−2).
Exercise 4
A car is
traveling on
a straight
road with a
velocity of
55 ft/sec at
𝑡 = 0.
For 0 ≤ 𝑡 ≤ 18 seconds, the car’s acceleration
𝑎(𝑡), in ft/sec2, is piecewise linear function defined
by the graph above.
On the time
interval 0 ≤ 𝑡 ≤ 18,
what is the car’s
absolute maximum
velocity, in ft/sec,
and at what time
does it occur.
Justify your
answer.
Total Distance
If an object moves along a straight line
according the position function 𝑠(𝑡) at time 𝑡,
then its velocity is 𝑣 𝑡 = 𝑠 ′ (𝑡) and its speed
is 𝑣 𝑡 .
𝑡2
𝑣 𝑡 𝑑𝑡 = 𝑉 𝑡2 − 𝑉 𝑡1
𝑡1
𝑉 is an antiderivative of 𝑣 𝑡
The total distance
traveled by the
object from
𝑡1 to 𝑡2 .
Total Distance
If an object moves along a straight line
according the position function 𝑠(𝑡) at time 𝑡,
then its velocity is 𝑣 𝑡 = 𝑠 ′ (𝑡) and its speed
is 𝑣 𝑡 .
𝑡2
𝑣 𝑡 𝑑𝑡 = 𝑉 𝑡2 − 𝑉 𝑡1
𝑡1
𝑉 is an antiderivative of 𝑣 𝑡
Exercise 5
A particle moves along a line so that its
velocity at time t is 𝑣 𝑡 = 𝑡 2 − 𝑡 − 6
(measured in meters per second). Find the
distance traveled during the time period
1 ≤ 𝑡 ≤ 4 seconds.
Exercise 6
Johanna jogs along a straight path. For 0 ≤ 𝑡 ≤ 40,
Johanna’s velocity is given by a differentiable
function 𝑣. Selected values of 𝑣(𝑡), where 𝑡 is
measured in minutes and 𝑣(𝑡) is measured in
meters per minute, are given in the table above.
Use the data in
the table to
estimate the
value of 𝑣′(16).
Exercise 7
Using correct units, explain the meaning of the
40
definite integral 0 𝑣 𝑡 𝑑𝑡 in the context of the
problem.
Exercise 8
40
0
Approximate the value of
𝑣 𝑡 𝑑𝑡 using a right
Riemann sum with the four subintervals indicated in
the table.
Objective 2
You will be able to numerically
approximate definite integrals
Approximation
Many functions do not
have a closed-form
No Closed-Form Antiderivative:
antiderivative, which
𝑓 𝑥 = 𝑥 cos 𝑥
means the antiderivative
cannot be generated from
𝑔 𝑥 = sin 𝑥 2
a finite combination of
elementary operations.
ℎ 𝑥 = 𝑒𝑥
For these, we need
methods of approximation.
2
Midpoint Rule
Before defining the definite integral, we used either
a left or a right Riemann sum to approximate the
area under a curve. A more accurate
approximation uses the midpoint of each
subinterval instead of the left or right endpoint.
Midpoint Rule
Let 𝑓 be continuous on 𝑎, 𝑏 where ∆𝑥 =
𝑛
𝑏
𝑓 𝑥 𝑑𝑥 ≈
𝑎
𝑖=1
𝑏−𝑎
.
𝑛
𝑥𝑖−1 + 𝑥𝑖
𝑓
∆𝑥
2
𝑥0 + 𝑥1
𝑥1 + 𝑥2
𝑥𝑛−1 + 𝑥𝑛
≈ ∆𝑥 𝑓
+𝑓
+ ⋯+ 𝑓
2
2
2
Midpoint Rule
Let 𝑓 be continuous on 𝑎, 𝑏 where ∆𝑥 =
𝑛
𝑏
𝑓 𝑥 𝑑𝑥 ≈
𝑎
𝑖=1
𝑏−𝑎
.
𝑛
𝑥𝑖−1 + 𝑥𝑖
𝑓
∆𝑥
2
𝑎 + 𝑥1
𝑥1 + 𝑥2
𝑥𝑛−1 + 𝑏
≈ ∆𝑥 𝑓
+𝑓
+⋯+𝑓
2
2
2
Exercise 9
Use a midpoint
sum with 4
subintervals to
approximate
2 2
𝑥 + 3 𝑑𝑥.
0
Compare your
approximation to
the actual area
under the curve.
Exercise 10
Hot water is dripping through a coffeemaker, filling
a large cup with coffee. The amount of coffee in the
cup at time 𝑡, 0 ≤ 𝑡 ≤ 6, is given by a differentiable
function 𝐶, where 𝑡 is measured in minutes. Select
values of 𝐶 𝑡 , measured in ounces, are given in
the table above.
Exercise 10
Use a midpoint sum with 3 subintervals of equal
length indicated by the data in the table to
1 6
approximate the value of 0 𝐶 𝑡 𝑑𝑡. Using correct
6
units, explain the meaning of
context of the problem.
1 6
𝐶
6 0
𝑡 𝑑𝑡 in the
Trapezoid Rule
The TI-84 uses what is
called the Gauss-Kronrod
Method to approximate a
definite integral, which is a
highly sophisticated
trapezoidal approximation.
Trapezoid Rule
Rather than
subdividing an area
into rectangles, we
could also use
trapezoids, which
is basically the
average of the left
and right Riemann
sums.
Trapezoid Rule
Area of First Trapezoid:
1
𝐴 = 𝑏1 + 𝑏2 ℎ
2
1
𝑏−𝑎
= 𝑓 𝑥0 + 𝑓 𝑥1
2
𝑛
1
= 𝑓 𝑥0 + 𝑓 𝑥1 ∆𝑥
2
=
1
𝑓 𝑥0 ∆𝑥 + 𝑓 𝑥1 ∆𝑥
2
Left
Right
Trapezoid Rule
Let 𝑓 be continuous on 𝑎, 𝑏 where ∆𝑥 =
𝑏
𝑓 𝑥 𝑑𝑥 ≈
𝑎
≈
1
2
∆𝑥
2
∆𝑥
≈
2
𝑛
𝑛
𝑓 𝑥𝑖−1 ∆𝑥 +
𝑖=1
𝑓 𝑥𝑖 ∆𝑥
𝑖=1
𝑛
𝑛
𝑓 𝑥𝑖−1 +
𝑖=1
𝑏−𝑎
.
𝑛
𝑓 𝑥𝑖
𝑖=1
𝑛
𝑓 𝑥𝑖−1 + 𝑓 𝑥𝑖
𝑖=1
≈
∆𝑥
𝑓 𝑥0 + 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛−1 + 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛
2
≈
∆𝑥
𝑓 𝑥0 + 2𝑓 𝑥1 + 2𝑓 𝑥2 + ⋯ + 2𝑓 𝑥𝑛−1 + 𝑓 𝑥𝑛
2
Trapezoid Rule
Let 𝑓 be continuous on 𝑎, 𝑏 where ∆𝑥 =
𝑏
𝑓 𝑥 𝑑𝑥 ≈
𝑎
≈
1
2
∆𝑥
2
∆𝑥
≈
2
𝑛
𝑛
𝑓 𝑥𝑖−1 ∆𝑥 +
𝑖=1
𝑓 𝑥𝑖 ∆𝑥
𝑖=1
𝑛
𝑛
𝑓 𝑥𝑖−1 +
𝑖=1
𝑏−𝑎
.
𝑛
𝑓 𝑥𝑖
𝑖=1
Coefficients of Sum:
𝑛
𝑓 𝑥𝑖−1 + 𝑓 𝑥𝑖
𝑖=1
≈
∆𝑥
𝑓 𝑥0 + 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛−1 + 𝑓 𝑥1 + 𝑓 𝑥2 + ⋯ + 𝑓 𝑥𝑛
2
≈
∆𝑥
𝑓 𝑎 + 2𝑓 𝑥1 + 2𝑓 𝑥2 + ⋯ + 2𝑓 𝑥𝑛−1 + 𝑓 𝑏
2
1 2 2 2 ⋯ 2 1
Exercise 11
Use a trapezoidal sum
with 4 subintervals
to approximate
2 2
𝑥 + 3 𝑑𝑥.
0
Compare your
approximation to the
actual area under
the curve and the
midpoint sum.
Exercise 12
Train 𝐴 runs back and forth on an east-west section
of railroad track. Train 𝐴’s velocity, measured in
meters per minute, is given by a differentiable
function 𝑣𝐴 𝑡 , where 𝑡 is measured in minutes.
Select values of 𝑣𝐴 𝑡 are given in the table above.
Exercise 12
At time 𝑡 = 2, train 𝐴’s position is 300 meters east
of the Origin Station, and the train is moving east.
Write an expression involving an integral that gives
the position of train 𝐴, in meters from the Origin
Station, at time 𝑡 = 12.
Use a trapezoidal
sum with three
subintervals
indicated by the
table to
approximate the
position of the train
at time 𝑡 = 12.
Total Change and Approximation
Objectives:
Assignment:
1. To use integrals of rates
of change to find total
change
• HW Worksheet: 1-10
2. To numerically
approximate definite
integrals
• HW Worksheet: 11-14