*
AP Calculus
Review
Approximating Definite
Integrals
Teacher Packet
AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not
involved in the production of this material.
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 1 of 14
The graph above represents the velocity of an object in motion, in ft/sec, for any time t, in
seconds.
To approximate distance traveled in 6 seconds, a Riemann sum may be used. Six
subdivisions would make maximum use of the data in the table. However, one could also
use 2 or 3 subdivisions. When subdivisions of equal width are to be used, calculate the
ending value − starting value
width of the subdivision ( Δt ) using the formula
. The
# of subdiv
process of the Riemann sums involves adding together the products of Δt ’s and
velocities. You are expected to use left side, right side, or midpoint values of the
velocity, as required by the problem.
A right side Riemann sum with three subdivisions would look like:
Distance = 2(33.44 + 20.32 + 16.64) = 140.8 feet
A left side Riemann sum with three subdivisions would look like:
Distance = 2(20 + 33.44 + 20.32) = 147.52 feet
A midpoint Riemann sum with three subdivisions would look like:
Distance = 2(32.29 + 27.95 + 15.05) = 150.58 feet
Note for midpoints: To find the velocity value, first find the midpoint of the time interval
and then use the corresponding velocity. The midpoint value is not found by finding the
midpoint of the velocity values. The midpoint method may not be applicable to all data
tables. On this table the midpoint method would not work with 6 subdivisions or 2
subdivisions.
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 2 of 14
Another method of approximating this distance is to use trapezoids. The formula for that
h(b1 + b2 )
area of a trapezoid is
. Subdivide the region from 0 to 6 and apply the area
2
formula for a trapezoid in each region.
Suppose you use 3 equal subdivisions, the approximation would be:
2(20 + 33.44)
2(33.44 + 20.32)
2(20.32 + 16.64)
+
+
= 144.16 feet
2
2
2
NOTE: These examples have all assumed subdivisions of equal width. In many data
problems the subdivisions cannot be equal in width. In those problems, the areas will
have to be calculated separately and then added together.
Example:
t (seconds)
v (feet/second)
0
34
8
65
11
67
17
43
Using three subdivisions you can use left or right Riemann sums (or a trapezoid sum).
Left = 8(34) + 3(65) + 6(67) = 869 feet
Right = 8(65) + 3(67) + 6(43) = 979 feet
Relative size of the approximations
If your function is increasing on the entire interval,
left sum < true value < right sum.
If your function is decreasing on the entire interval,
right sum < true value < left sum.
If your function is concave down on the entire interval,
trapezoid sum < true value < midpoint sum.
If your function is concave up on the entire interval,
midpoint sum < true value < trapezoid sum.
While this information is relating to a problem of finding distance, you can use all of
these methods to approximate any definite integral.
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 3 of 14
Sample Problems
Multiple Choice – No Calculator
1.
If
∫
11
x dx is approximated with 5 midpoint rectangles of equal width, then the
1
approximation is
(A)
2 + 2 + 6 + 2 2 + 10
(B) 2
(
2 + 2 + 6 + 2 2 + 10
(
(D) 2
(
3 + 5 + 7 + 3 + 11
(C) 2 1 + 3 + 5 + 7 + 3
)
)
)
⎛ 1+ 3
3+ 5
5+ 7
7 + 3 3 + 11 ⎞
+
+
+
+
(E) 2 ⎜⎜
⎟
2
2
2
2
2 ⎟⎠
⎝
2.
The graph of f is given. An estimate of
(A) 16
(B) 19
(C) 80
®
∫
5
0
f ( x) dx is
(D) 95
(E) 100
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 4 of 14
3.
Approximate
(A) 52
∫
14
2
2x dx using a left Riemann sum and 4 equal subdivisions.
(B) 76
(C) 156
(D) 192
(E) 228
4.
Time
(milliseconds)
Light output
(millions of
lumens)
0
5
10
15
20
25
30
0
.2
.6
2.6
4.2
3
1.8
The data in the table represents the rate of light output of a flash bulb at a given
time. Using a midpoint approximation with 3 equal subdivisions estimate the
total light output of the bulb, measured in million lumen-milliseconds.
(A) 5.8
5.
(B) 48
(C) 57
(D) 58
(E) 66
The table below gives data points for a continuous function f on [2, 10].
x
f (x)
2
3
5
1
7
5
8
3
10
5
Using subdivisions [2, 5], [5, 8] and [8, 10], what is the trapezoid approximation
of
∫
10
2
(A) 18
f ( x) dx ?
(B) 20
(C) 22
®
(D) 24
(E) 40
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 5 of 14
6.
The table below gives data points for a continuous function f on [3, 12].
3
2
x
f (x)
5
4
8
6
12
4
What is the absolute value of the difference between a right sum and a left sum
when approximating
(A) 0
7.
Estimate
∫
12
f ( x) dx with 3 intervals?
3
(B) 2
π
∫π
(C) 4
(D) 6
(E) 7
sin( x) dx using a right Riemann sum with 2 equal subdivisions.
2
(A) π (1 + 0)
(D)
π⎛ 2
⎞
− 1⎟⎟
⎜⎜
4⎝ 2
⎠
(B)
(E)
®
π⎛
2 ⎞
− 1⎟⎟
⎜⎜ −
4⎝ 2
⎠
(C)
π⎛ 2
⎞
+ 1⎟⎟
⎜⎜
4⎝ 2
⎠
π⎛ 2
⎞
+ 0 ⎟⎟
⎜⎜
4⎝ 2
⎠
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 6 of 14
Free Response 1 – No Calculator
An object moves in a straight line during 0 < t < 12 seconds. The velocity (m/sec) and
acceleration (m/sec2) are differentiable functions. The table shows selected values of the
functions.
t (sec)
v (m/sec)
a (m/sec2)
0
0
1
2
4
3
5
8
2
6
6
4
10
10
1
12
30
3
(a)
Using a trapezoid sum and 5 intervals estimate
(b)
Using correct units explain the meaning of
(c)
Using a left sum and 5 intervals approximate
answer. What is the true value of
(d)
∫
12
0
Explain why acceleration must equal
®
∫
12
0
v(t ) dt .
1 12
v(t ) dt .
12 ∫ 0
∫
12
0
a(t ) dt . Include units in your
a(t ) dt ?
1
for some value on [2, 6].
2
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 7 of 14
Multiple Choice – Calculator Allowed
1.
A radar gun was used to record the velocity (m/sec) of a runner. Data points for
that event are recorded in the table. Use a trapezoid sum to approximate distance
covered in 8 seconds.
t (sec)
v (m/sec)
2.
0
0
4.3
10.83
6.5
12.46
(A) 42.516
(B) 65.239
(D) 130.477
(E) 260.954
8
9.32
(C) 87.961
What is the absolute value of the difference between the exact value for
∫
7
−1
(3 x − 1) dx and a midpoint approximation with 4 equal subdivisions?
(A) 0
(B) 22
(C) 40
(D) 64
(E) 88
3.
The graph represents a continuous function f on [0, 3]. If L3, R3, T3 and M3
represent the approximations of
∫
3
0
f ( x) dx - left sum, right sum, trapezoid sum,
and midpoint sum, respectively. Which of the following is not true?
(A) L3 < R3
(B) T3 < R3
(D) M3 < L3
(E) M3 < R3
®
(C) T3 < M3
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 8 of 14
Free Response – Calculator Allowed
1.
Let R (t ) represent the rate of change of the value of a stock, in dollars per day
over a 12 day period.
t (days)
R (t )
($/day)
(a)
0
2
5
9
11
12
10
22
15.77
13.29
12.70
12.48
Using a right Riemann sum and 5 intervals estimate
∫
12
0
R(t ) dt . Explain
the meaning of this integral using correct units.
(b)
What is the average rate of change in R (t ) over [0, 12]?
(c)
Using your work from part (a), what is the average value of R (t ) over
[0, 12]? Include units.
(d)
30t
+ 10 be a model for the rate of
1+ t2
change in the value of the stock on the interval [0, 12]. For what value(s)
of t does the instantaneous rate of change in Q equal the average rate
change of Q? Show the equation that leads to the answer.
Let the function defined by Q(t) =
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 9 of 14
Free Response – Calculator Allowed
2.
200 x + 10
be a function that models the rate, in gallons per hour, at
2 x 2 + 15
which a water tank is being filled over at 24-hour period.
20 x 2 − 50 x + 160
Let E (t ) =
be a function that models the rate, in gallons per
2 x 2 + 15
hour, at which a water tank is being emptied over the same 24-hour period.
Assume the tank contained 50 gallons at t = 0.
Let F (t ) =
(a) Set-up the approximation for
∫
24
0
F (t ) dt using midpoint sums and 4 equal
subdivisions.
(b) Find F (1) − E (1) . Explain what the answer tells you about the water in the
tank. Include correct units.
(c) Evaluate
∫
1
0
( F (t ) − E (t )) dt . Explain what the answer tells you about the
water in the tank. Include correct units.
(d) On what interval(s) is the level of water in the tank rising? Explain your
answer.
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 10 of 14
Key
No Calculator
MC
1. B
2. C
3. C
4. D
5. B
6. B
7. E
Calculator Allowed
MC
1. B
2. A
3. D
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 11 of 14
Free Response 1 – No Calculator
An object moves in a straight line during 0 < t < 12 seconds. The velocity (m/sec) and acceleration
(m/sec2) are differentiable functions. The table shows selected values of the functions.
t (sec)
v (m/sec)
a (m/sec2)
0
0
1
2
4
3
5
8
2
6
6
4
10
10
1
12
30
3
(a)
Using a trapezoid sum and 5 intervals estimate
(b)
Using correct units explain the meaning of
(c)
Using a left sum and 5 intervals approximate
the true value of
(d)
(a)
∫
12
0
∫
12
0
v(t ) dt .
1 12
v (t ) dt .
12 ∫ 0
∫
12
0
a (t ) dt . Include units in your answer. What is
a(t ) dt ?
Explain why acceleration must equal
1
for some value on [2, 6].
2
1
( 2 )( 0 + 4 ) = 4
2
1
( 3)( 4 + 8) = 18
2
1
(1)(8 + 6 ) = 7
2
1
( 4 )( 6 + 10 ) = 32
2
1
( 2 )(10 + 30 ) = 40
2
2 pts: 1 pt trapezoid set up
1 pt answer
The Trapezoid Sum equals 101 meters.
(b) The average value of the velocity (the average
velocity) function over the 12 hour interval
in meters per second.
1 pt correct explanation
(c) LRS = 2(1) + 3(3) + 1(2) + 4(4) + 2(1) = 31
The LRS is 31 meters per second
4 pts: 1 pt LRS
1 pt v(12) − v(0)
∫
12
0
a(t ) dt = v(12) − v(0) = 30
1 pt LRS answer
The actual value of the above integral is
30 meters per second.
(d) Because the functions are differentiable, the
Mean Value Theorem applies.
v(6) − v(2) 1
=
6−2
2
®
1 pt v(12) − v(0) answer
2 pts: 1 pt applying MVT
1 pt
v(6) − v(2) 1
=
6−2
2
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 12 of 14
Free Response – Calculator Allowed
1.
Let R (t ) represent the rate of change of the value of a stock, in dollars per day over a 12 day
period.
t (days)
R (t )
($/day)
(a)
(b)
(c)
(d)
0
2
5
9
11
12
10
22
15.77
13.29
12.70
12.48
Using a right Riemann sum and 5 intervals estimate
12
0
R (t ) dt . Explain the meaning of
this integral using correct units.
What is the average rate of change in R (t ) over [0, 12]?
Using your work from part (a), what is the average value of R (t ) over [0, 12]? Include
units.
30t
Let the function defined by Q(t) =
+ 10 be a model for the rate of change in the
1+ t2
value of the stock on the interval [0, 12]. For what value(s) of t does the instantaneous
rate of change in Q equal the average rate change of Q? Show the equation that leads to
the answer.
(a) RRS = 2(22) + 3(15.77) + 4(13.29) + 2(12.70) + 1(12.48)
= $182.35
The stock value increased by $182.35 over
the twelve day interval.
(b)
∫
R(12) − R(0) 12.48 − 10
=
= 0.2066...
12 − 0
12
3 pts: 1 pt set up
1 pt answer
1 pt explanation
R (12) − R(0)
12
2 pts: 1 pt
1 pt answer
(c)
1 12
R(t ) dt = $15.1958 per day
12 ∫ 0
2 pts: 1 pt
1
12
of answer in
1 pt for units $ per day
(d)
Q(12) − Q(0)
= Q′(t )
12 − 0
t = 0.986
2 pts: 1 pt equation
1 pt answer
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 13 of 14
Free Response – Calculator Allowed
200 x + 10
be a function that models the rate, in gallons per hour, at which a water
2 x 2 + 15
tank is being filled over at 24-hour period.
20 x 2 − 50 x + 160
Let E (t ) =
be a function that models the rate, in gallons per hour, at which a
2 x 2 + 15
water tank is being emptied over the same 24-hour period.
Assume the tank contained 50 gallons at t = 0.
Let F (t ) =
2.
(a) Set-up the approximation for
(b)
∫
24
0
F (t ) dt using midpoint sums and 4 equal subdivisions.
Find F (1) − E (1) . Explain what the answer tells you about the water in the tank. Include
correct units.
(c) Evaluate
∫
1
0
( F (t ) − E (t )) dt . Explain what the answer tells you about the water in the tank.
Include correct units.
(d) On what interval(s) is the level of water in the tank rising? Explain your answer.
(a) 6( F (3) + F (9) + F (15) + F (21))
(b) F (1) − E (1) = 4.7058
At t = 1 hour, the amount of water is
increasing at a rate of 4.705 or 4.706
gallons per hour.
(c)
∫
1
0
( F (t ) − E (t )) dt = −2.177
2 pts: 1 pt Δt = 6
1 pt F (t ) s
2 pts: 1 pt answer
1 pt explanation
2 pts: 1 pt answer
During the first hour, the amount of water in
the tank decreases by 2.1773 gallons
-or- The tank holds 47.8227 gallons and the
end of the first hour.
(d) When F (t ) − E (t ) > 0 , the water level rises.
This occurs over the interval (0.6319, 11.8688) .
1 pt explanation
1 pts: 1 pt correct interval
1 pt explanation
1 pt units on (b) and (c)
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Approximating Definite Integrals
(Riemann and Trapezoid Sums)
Page 14 of 14
AP Calculus Exam Connections
The list below identifies free response questions that have been previously asked on the
topic of Approximating Definite Integrals (Riemann and Trapezoid Sums). These
questions are available from the CollegeBoard and can be downloaded free of charge
from AP Central.
http://apcentral.collegeboard.com.
Free Response Questions
2004 Form B
2006 Form B
2006
2007
®
AB Question 3
AB Question 6
AB Question 4
AB Question 5
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