Name:
Date:
Block:
Integration Revision 1
Non–Calculator
4
1) Write the definite integral as a limit sum.
 (2 x
3
 4)dx
1
2
n

3i 
 3 
2) Express as an integral: lim     2  4    3
n 
n
i 1  n  

 
8
3) Express as a limit sum:
 (5 x
2
 3 x  2) dx
3
7
4) Use the trapezoid rule with 3 equal intervals to approximate  (2 x 2  3x  1)dx
1
x
5) Given F ( x)   sin 3tdt , find F ( x) .

6
6) Find the average value of the function f ( x)  3x 2  2 over the interval [1,1] .
8
7) Use the data from the table and trapezoids to estimate
 f ( x)dx .
0
x
0
2
4
6
8
f ( x)
8
–5
7
6
–1
8) The acceleration of a particle moving along the x–axis at time t is given by a(t )  2t  7 . If the initial
velocity is 10 and the initial position is −1, then the particle is changing direction at
(A) t = 2
(B) t = 5
(C) t = 10
(D) t =2, t = 5
(E) t = 2, t = 5, and t = 10
9) The average value of the function f ( x)   x  1 on the interval from x = 1 to x = 5 is
2
16
3
16
(B)
3
64
(C)
3
66
3
256
(E)
3
(A) 
2
10)
1
x
2
(D)
dx =
1
(A)
3
8
(B) 


3
4
(C)
1
2
(D) 
3
2
(E) 0
v(t)
Kilometers per hour








t









Hours
11) A car’s velocity is shown in the graph above. Which of the following gives the total distance the car
travelled form t = 0 to t = 8 (in kilometers)?
(A) 0
(B) 240
(C) 390
(D) 760
(E) 940
x2
12) If f ( x) 
 t
3

 t dt , then f (1) =
0
(A)
13)

x
0
3
(B) 4
(C) −4
(D) 1
(E) 2
(B) cos x
(C) −cos x
(D) tan x
(E) tan x – 1
sec2 t dt =
(A) sin x
Free Response – Non–Calculator
t
(seconds)
v(t)
(feet per second)
0
10
20
30
40
50
60
70
80
5
14
22
29
35
40
44
47
49
1) Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time
t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0  t  80
seconds, as shown in the table above.
a) Find the average acceleration of rocket A over the time interval 0  t  30 seconds. Indicate units
of measure.
60
b) Using correct units, explain the meaning of
 v(t )dt
in terms of the rocket’s flight.
30
80
c) Use a midpoint Riemann sum with 3 subintervals of equal length to approximate
 v(t )dt .
20
d) Rocket B is launched upward with an acceleration of a(t )  4t  5 feet per second per second.
At time t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 226 feet
per second. Find an equation for the velocity of rocket B at any time t.
e) Write an expression that represents the average acceleration of rocket B over the time interval
0  t  60 seconds.
Free Response – Calculator Active
2) The temperature during the last Steelers game in Pittsburg was given by the equation
H 
T ( H )  A  B cos 
,
 2 
where T is the temperature in degrees Fahrenheit and H is the number of hours from midnight  0  H  24  .
a) People began to show up to tailgate at 3 a.m., when the temperature was 20F. Seven hours and
many bratwursts later, the temperature fell to 5F. Find A and B.
b) Find the average temperature for the first 9 hours.
c) Use the Trapezoid Rule with 4 equal subdivisions to estimate
Riemann sum.

6
4
T ( H )dH . Be sure to show your
d) Find an expression for the rate that the temperature is changing with respect to H.
1)
t
(seconds)
v(t)
(feet per second)
0
10
20
30
40
50
60
70
80
5
14
22
29
35
40
44
47
49
Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t = 0
seconds. The velocity of the rocket is recorded for selected values of t over the interval 0  t  80 seconds,
as shown in the table above.
a) Find the average acceleration of rocket A over the time interval 0  t  30 seconds. Indicate units of
measure.
1 Answer
v(30)  v(0) 31  15 16 8



ft / sec
1 Units
30  0
30
30 15
b) Using correct units, explain the meaning of

60
30

60
30
v (t ) dt in terms of the rocket’s flight.
v (t ) dt represents the total distance travelled by the rocket from t = 30 to t = 60 seconds.
1
1
c) Use a midpoint Riemann sum with 3 subintervals of equal length to approximate
80  20
v(30)  v(50)  v(70)
3
M  20[31  50  59]
M
1 Setup
M  2800
1 M  2800

80
20
v(t )dt .
d) Rocket B is launched upward with an acceleration of a(t )  4t  5 feet per second per second.
At time t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 226 feet per
second. Find an equation for the velocity of rocket B at any time t.
v (t )   a (t ) dt
v(0)  2(0) 2  5(0)  C  226
v (t )   4t  5dt
 C  226
1 C = 226
v(t )  2t 2  5t  226
v (t )  2t 2  5t  C
1 v(t)
e) Write an expression that represents the average acceleration of rocket B over the time interval
0  t  60 seconds.
1
1
a (t )dt
4t  5dt
OR

60  0
60  0 
1
1
OR
1
1
2) The temperature during the last Steelers game in Pittsburg was given by the equation
H 
T ( H )  A  B cos 
,
 2 
where T is the temperature in degrees Fahrenheit and H is the number of hours from midnight  0  H  24  .
a) People began to show up to tailgate at 3 a.m., when the temperature was 20F. Seven hours and many
bratwursts later, the temperature fell to 5F. Find A and B.
 3 
 10 
20  A  B cos  
5  20  B cos 

H 
 2 
 2 
T ( H )  20  15cos 

 2 
20  A  B(0)
15  B cos(5 )
20  A
15  B(1)
15  B
1 Equation for T(H)
b) Find the average temperature for the first 9 hours.
1 9
 9 
20  15cos 
  21.06 F

0
90
 2 
1 9
T ( H )  21.06 F
9  0 0
1
1
90
9
1
 T ( H ) or equivalent expression
1
21.06
0
c) Use the Trapezoid Rule with 4 equal subdivisions to estimate

6
4
T ( H )dH . Be sure to show your
Riemann sum.
64
 f (4)  2 f (4.5)  2 f (5)  2 f (5.5)  f (6)
2(4)
1
T  35  2(30.607  2(20)  2(9.3934)  5
4
T  40.002
T
1
Setup
1
Solution
Note: 0/1 if uses equal sign
d) Find an expression for the rate that the temperature is changing with respect to H.
H 
T ( H )  20  15cos 

 2 
15
H 
T ( H ) 
sin 

2
 2 
1 T ( H )