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Period________________
Practice TEST: AB/BC Methods of Integration (through AB topics)
Part I: MULTIPLE CHOICE (USE CAPITAL LETTERS)
Use a finite approximation to estimate the area of the region enclosed between the graph of f and the x-axis for a
b.
1) f(x) = x 2 , a = 3, b = 7
1)
Use LRAM with four rectangles of equal width.
A) 117
B) 86
C) 105
2) f(x) = x 2 , a = 3, b = 7
Use RRAM with four rectangles of equal width.
A) 117
B) 126
D) 126
2)
C) 105
D) 86
Estimate the value of the quantity.
3) The table shows the velocity of a remote controlled race car moving along a dirt path for 8
seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with
left-end point values.
Time Velocity
(sec) (in./sec)
0
0
1
10
2
20
3
16
4
26
5
29
6
31
7
12
8
5
A) 288 in.
B) 144 in.
C) 134 in.
D) 149 in.
Use the Trapezoidal Rule to estimate the integral.
4
f(x) dx
4)
1
x
1
2
3
4
f(x) 3.6 7.3 10.6 12.6
A) 33.3
4)
B) 36.6
C) 26
D) 28.5
Graph the integrand and use areas to evaluate the integral.
5
25 - x 2 dx
5)
5)
-5
A) 5
3)
B) 25
C)
1
25
2
D) 25
x
6)
5
(5 - x ) dx
6)
-5
A) 50
B) 25
C)
25
2
D) 75
Express the desired quantity as a definite integral and evaluate the integral.
7) Find the output of a pump that produces 18 gallons per minute during the first 5 hours of its
operation.
300
300
18 dt , 90 gal
18 dt , 5400 gal
A)
B)
0
C)
18
0
300 dt , 5400 gal
D)
0
5
18 dt , 90 gal
0
Solve the problem.
6
8) Suppose that
A) 4; -4
4
f(x) dx = -4. Find
4
B) 0; 4
f(x) dx and
4
9) Suppose that h is continuous and that
4
f(x) dx .
6
C) -4; 4
4
-2
8)
D) 0; -4
8
h(x) dx = 2 and
h(x) dx = -8. Find
4
-2
and
7)
8
h(t) dt
9)
-2
h(t) dt .
8
A) 6; -6
B) 10; -10
C) -6; 6
D) -10; 10
USE NINT to find the average value of the function on the interval. At what point in the interval does the function
assume its average value?
10) y = -4x 2 - 1, [0, 3.87298335]
10)
A) -61, at x = 3.87298335
C) 61, at x = 3.87298335
B) 21, at x = 2.23606798
D) -21, at x = 2.23606798
2
Find the average value of the function without integrating, by appealing to the geometry region between the graph
and the x-axis.
11) f(x) = x + 7, -7 x -1
11)
-x + 5, -1 < x 5
A) 2
B) 6
Evaluate the definite integral.
e
2
dx
12)
x
1
A) - 1e2
C) 3
x 10
14)
B) -2
C) 2
C)
98
D) 0
D)
2
cos t dt
0
A) cos (x 5 ) - 1
6 ln x
15)
7
2
12)
Find the average value over the given interval.
13) y = 7 sin x; [0, ]
7
14
A)
B)
Find dy/dx.
D)
14)
B) sin (x 5 )
C) cos (x 5 )
D) 10x 9 cos (x 5 )
et dt
15)
0
A) 6x 5
13)
B) x 6
C) x 6 - 1
3
D)
6 ex 6
x
x
Construct a function of the form y =
f(t) dt + C that satisfies the given conditions.
a
16)
dy
= csc x, and y = -9 when x = 4
dx
x
A) y = 4
x
C) y =
16)
csc t cot t dt - 9
4
B) y =
x
csc t dt + 4
D) y =
x
csc t dt - 9
csc t dt - 9
4
-9
Evaluate the integral.
-1
4 x dx
17)
17)
3
A)
-63
4 ln 4
B)
65
4 ln 4
C)
-257
4 ln 4
D)
Find the total area of the region between the curve and the x-axis.
18) y = x 2 - 6x + 9; 2 x 4
A)
7
3
B)
1
3
C)
4
3
D)
-255
4 ln 4
2
3
Find the area of the shaded region.
19)
A)
9
4
18)
19)
B)
33
4
C)
4
41
4
D)
17
4
20)
20)
y=2
y = csc2 x
A)
+2
B)
C)
Use NINT to solve the problem.
10
1
dx.
21) Evaluate
4 + 3 cos x
0
A) 0.026
B)
D) 2
21)
0.022
Solve the problem.
x+1
22) Find the linearization of f(x) = 9 +
1
A) 9x + 1
-2
C)
tan
3.107
D)
4.064
t
dt at x = 0.
4
C) 2x + 9
B) 9
22)
D) x + 9
Evaluate the integral.
x
dx
23)
3x 2 - 6
A)
x
+C
(3x 2 - 6)2
C) -
24)
25)
23)
7x 2
B)
6
+C
(3x 2 - 6)2
4
1
ln 3x 2 - 6 + C
6
D) x ln 3x 2 - 6 + C
4 + 3x 3 dx
24)
A)
28
(4 + 3x 3 ) 5/4 + C
5
B) -
14
-3/4 +
(4 + 3x 3 )
C
3
C)
28
(4 + 3x 3 ) 5/4 + C
45
D) 7(4 + 3x 3 ) 5/4 + C
ln9 x
dx
x
A) ln10x + C
25)
B)
ln8 x
+C
8
C)
5
ln10x
+C
10x
D)
ln10x
+C
10
26)
dx
26)
x ln x 6
A) ln (ln x 6 ) + C
27)
28)
1
ln (ln x 6 ) + C
6
C) ln x 6 + C
1
x
+C
tan-1
10
5
B)
1
x
+C
tan-1
25
25
C)
1
x
+C
tan-1
25
5
D)
1
x
+C
tan-1
5
5
dx
sin2 (5x)
28)
1
cot 5x + C
5
B)
1
cot 5x + C
5
D) - cot 5x csc 5x + C
29)
1
sec6 x + C
6
C) - 6 tan6 x + C
B)
1
tan6 x + C
6
D)
1
tan5 x sec x + C
5
1
dx
cot (9x - 5)
A) C)
31)
1
csc3 5x + C
3
tan5 x sec2 x dx
A)
C)
B) - ln cos (9x - 5) + C
1
ln cos (9x - 5) + C
9
(3 + cos t)3
A)
30)
1
ln cos (9x - 5) + C
9
sin t
D) ln sin (9x - 5) + C
31)
dt
2
(3 + cos t)2
1
1
ln x 6 + C
6
27)
A)
C) -
30)
D)
dx
x 2 + 25
A)
29)
B)
(3 + cos t)2
+C
B)
+C
D)
6
1
4(3 + cos t)4
1
2(3 + cos t)2
+C
+C
32)
cos(4 + 5)
d
2
sin (4 + 5)
A)
1
+C
4 sin(4 + 5)
C) -
33)
32)
B) -
1
+C
4 sin(4 + 5)
D)
cos(4 + 5)
+C
4 sin(4 + 5)
1
+C
sin(4 + 5)
tan2 4x dx
A)
33)
1
tan 4x sec 4x - x + C
4
B) x -
C) 4 tan 4x - x + C
D)
1
tan 4x + C
4
1
tan 4x - x + C
4
Use the given trig identity to set up a u-substitution and then evaluate the indefinite integral.
34)
(cos4 6x - sin4 6x) dx , cos 12x = cos2 6x - sin2 6x
A) C)
1
cos 12x + C
12
1
1
cos3 12x - sin3 12x + C
3
3
7
34)
B)
1
1
cos5 6x - sin5 6x + C
5
5
D)
1
sin 12x + C
12