Math 1206
Common Final Exam
Fall 2012
Form A
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1. Which one of the listed table integral formulas
u√a2 + u2
√
u2
a2 √
A:
+C
du = − ln u + a2 + u2 +
2
2
a2 + u 2
√
Z
1
a2 + u 2
√
B:
du = −
+C
a2 u
u2 a2 + u2
!
√
Z
1
a + a2 + u2
1
√
+C
C:
du = − ln
a
u
u a2 + u2
Z
can be applied to the integral below after an appropriate substitution?
Z
1
q
dx
1 2
1+ x
(1) A
(2) B
(3) C
(2) 1
(3)
(4) None of them
sin(t2 )
.
t→0 1 − cos(t)
2. Evaluate lim
(1) Does not exist
−1
2
(4) 2
3. An electronic device in a gasoline pump measures the pumping rate V 0 (t) in gallons per second. The
table below lists the output of this device when you were fueling your car. Use Simpson’s rule to
estimate the amount of gasoline V that you have pumped in your car.
t
0
1
2
3
4
5
6
0
V (t) 0.2 0.3 0.2 0.15 0.15 0.1 0.1
(1)
1
(0.2
3
+ 4 · 0.3 + 2 · 0.2 + 4 · 0.15 + 2 · 0.15 + 4 · 0.1 + 0.1)
(2) 1 · ( 12 · 0.2 + 0.3 + 0.2 + 0.15 + 0.15 + 0.1 + 12 · 0.1)
(3)
1
(0
3
(4)
1
(0.2
2
· 0.2 + 1 · 0.3 + 2 · 0.2 + 3 · 0.15 + 4 · 0.15 + 5 · 0.1 + 6 · 0.1)
+ 2 · 0.3 + 2 · 0.2 + 2 · 0.15 + 2 · 0.15 + 2 · 0.1 + 0.1)
Z
4. Evaluate the integral
∞
2
xe−x dx.
0
(1)
1
e
(2) 1
(3)
1
2
(4)
Integral
diverges
√
5. Find the curve y = f (x) that passes through the point (4, 9) and whose slope at each point is 3 x.
(1) f (x) = 2x3/2 − 7
(2) f (x) = 2x3/2 − 9
(3) f (x) = 3x3/2 − 15
√
(4) f (x) = 3 x − 6
6. Suppose f (x) = ex + c where c is a constant. Suppose also that the average value of f over [0, ln 2]
1
is
. Then,
ln 2
1
1
1
−
(1) c = −
(2) c = −
2
(ln 2)
ln 2
ln 2
(3) c =
1
2
−
2
(ln 2)
ln 2
Z
7. Evaluate
1
(1)
2
3
3
(4) c = 0
(x + 1)(x − 1)
dx.
x2
(2)
4
3
(3) 2
(4)
8
3
dy
if y =
8. Find
dx
Z
1
cos(x)
1
dt.
ln(t)
(1) ln ln(cos(x))
(2) −
sin(x)
ln cos(x)
(3)
sin(x)
ln cos(x)
(4) −
1
ln cos(x)
9. Suppose the curves f (x) and g(x) only intersect at x = −2, x = 0, and x = 2. It is known that
Z 0
Z 2
(f (x) − g(x)) dx = −3
and
(f (x) − g(x)) dx = 5.
−2
0
What is the total area of the regions bounded by f (x) and g(x)?
5−3
(1) 2
(2)
(3) 8
2
(4)
5+3
2
10. A tank with the shape of an inverted right circular cone (the vertex is down) is filled with milk (weighing
64.5 lb/ft3 ) to a depth of one half its height. If the height is 20 ft. and its diameter is 8 ft., find the
work done by pumping all of the milk to the top of the tank. Assume the origin is at the vertex.
Z 10
(1) 64.5π
(5y)2 (20 − y) dy
0
d=8 ft.
Z
10
(2) 64.5π
y 2
5
20 ft.
0
Z
20
(3) 64.5π
y 2
5
10
Z
20
(4) 64.5π
y 2
10
5
(20 − y) dy
(20 − y) dy
(y − 10) dy
11. Use √
the shell method to find the volume of the solid generated by a region bounded by the graphs
y = x, x = 4 and y = 0 revolved around y = 2.
Z 2
Z 2
2
(1) 2π
(2) 2π
(y)(4 − y) dy
(2 − y)(4 − y 2 ) dy
0
Z
(3) 2π
0
0
4
4 − (2 −
√
2
x)
Z
dx
(4) π
4
(2 −
0
√
x)2 dx
12. Calculate the arc length of the graph y = sin(x) on the interval [0, π].
Z π
Z π/2 q
2
1 + cos2 (x) dx
(1)
(2)
1 + sin2 (x) dx
0
0
π
Z
p
1 + cos2 (x) dx
(3)
(4)
0
Z
π
r
1+
0
sin2 (x)
dx
x2
13. Let region R be bounded by x = 2y + 5 and x = y 2 + 2. The moment with respect to the y-axis of
region R with constant density δ = 2 is:
R3
Z 3
y(3 + 2y − y 2 ) dy
−1
(1)
(2) 2
y(3 + 2y − y 2 )dy
R3
2
3 + 2y − y dy
−1
−1
R3
(3)
−1
Z
14. Evaluate
(2y + 5)2 − (y 2 + 2)2 dy
R3
2 −1 3 + 2y − y 2 dy
Z
3
(2y + 5)2 − (y 2 + 2)2 dy
(4)
−1
25x3 + 16x − 8
dx.
25x2 + 16
(1)
x2
− 8x + C
2
(3)
x2
− ln(25x2 + 16) + C
2
(2)
x2 2
− arctan
2
5
(4)
x2 8
− arctan
2
5
5x
4
5x
4
+C
+C
15. An appropriate substitution would transform the integral
Z
10
√
dx
x3 x2 − 25
into which of the following?
Z
2
(1)
cos2 θ dθ
(2)
25
2
25
Z
2
sec θ dθ
(3)
2
25
Z
3
cos θ dθ
(4)
2
25
Z
tan θ
dθ
sec3 θ