Finals Math 2414 Litong Name __________________________________________________ Test No, ____________
Solve the problem.
Use the substitution formula to evaluate the integral.
8) It takes a force of 12,000 lb to compress a
π/2
1)
cot x csc5 x dx
spring from its free height of 15 in. to its fully
compressed height of 10 in. How much work
π/6
does it take to compress the spring the first
inch?
Find the area enclosed by the given curves.
∫
2) x = 2y2 - 3 and x = y2 + 6
9) A fisherman is about to reel in a 16-lb fish
located 14 ft directly below him. If the fishing
line weighs 1 oz per foot, how much work will
it take to reel in the fish? Round your answer
to the nearest tenth, if necessary.
Answer each question appropriately.
3) Which of the following integrals, if any,
calculates the area of the shaded region?
5
(-2, 4)
y
Find the center of mass of a thin plate of constant density
covering the given region.
10) The region enclosed by the parabolas y = - x2
(2, 4)
4
3
2
+ 72 and y = x2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
Find the fluid force exerted against the vertically
submerged flat surface depicted in the diagram. Assume
arbitrary units, and call the weight-density of the fluid w.
11)
-1
-2
-3
-4
-5
A)
∫
2
-2
0
C)
∫
-2
4x dx
-4x dx
B)
∫
-2
4
D)
∫
9
0
4x dx
9
9
-4x dx
-4
Find the formula for df-1 /dx.
12) f(x) = 27x3
Find the volume of the solid generated by revolving the
region about the given line.
4) The region in the first quadrant bounded
above by the line y = 3, below by the curve y =
3x, and on the left by the y-axis, about the
line y = 3
Solve the problem.
13)
If f(x) is one-to-one, is g(x) = f(-x) also
one-to-one? Explain.
Use the shell method to find the volume of the solid
generated by revolving the region bounded by the given
curves and lines about the x-axis.
5) y = 4 x , y = 4
Find the derivative of y with respect to x, t, or θ, as
appropriate.
ln x
14) y = x7
Find the length of the curve.
6) x = 6 sin t + 6t, y = 6cos t, 0 ≤ t ≤ π
15) y = ln ln 5x
Evaluate the integral.
2 dx
16)
1 + 3x
Find the area of the surface generated by revolving the
curves about the indicated axis.
7) x = sin t, y = 4 + cos t, 0 ≤ t ≤ 2π; x-axis
∫
1
17)
∫
dx
x 2 + 9 ln x
Solve the initial value problem.
dy
-6
29)
= , y(1) = -5
dx
1 - x2
Use logarithmic differentiation to find the derivative of y.
x
18) y = x + 5
^
Use lʹHopitalʹs rule to find the limit.
sin x
30) lim
4
x→0 + x
Find the derivative of y with respect to x, t, or θ, as
appropriate.
19) y = ln (7θe-θ)
Find the derivative of y.
31) y = sinh2 9x
20) y = sin e-θ6
Find the derivative of y with respect to the appropriate
variable.
32) y = 10 tanh-1 (cos x)
Evaluate the integral.
9e(9 sin 5x)
dx
21)
sec 5x
∫
22)
∫
Evaluate the integral.
e1/x
dx
2x2
Find the derivative of y with respect to the independent
variable.
23) y = (ln 5θ)π
Use logarithmic differentiation to find the derivative of y
with respect to the independent variable.
24) y = (cos x)x
Find the derivative of y with respect to x.
2x
25) y = tan-1 5
28)
∫
7 + 10x
4 + 49x2
34)
∫ e2x cos 6x dx
35)
∫ x4 ln 8x dx
36)
∫ csc3 4t dt
37)
∫
π/2
cos 8t cos 7t dt
Integrate the function.
dx
38)
(x2 + 4)3/2
∫
∫
∫
∫ 2 sinh (4x- ln 3) dx
0
Evaluate the integral.
7 - 5x
26)
dx
64 - 49x2
27)
33)
39)
∫
40)
∫
dx
dx
-x2 - 6x - 8
2
dx
, x > 4
x2 x2 - 16
1
t2 9 - t2
dt
Express the integrand as a sum of partial fractions and
evaluate the integral.
7
3x dx
41)
(x - 1)3
3
Graph the parabola or ellipse. Include the directrix that
corresponds to the focus at the origin.
12
50) r = 4 + 4 cos θ
∫
6
Evaluate the integral by first performing long division on
the integrand and then writing the proper fraction as a
sum of partial fractions.
7x3 + 7x2 + 8
dx
42)
x2 + x
4
∫
2
-6
Evaluate the improper integral or state that it is divergent.
∞
2dx
43)
36 + x2
0
-4
-2
2
-2
∫
-4
-6
Evaluate the improper integral.
7
dx
44)
49 - x2
0
∫
Find the limit of the sequence if it converges; otherwise
indicate divergence.
6 + (-1)n
45) a n = 6
Determine if the series converges or diverges; if the series
converges, find its sum.
∞
4
46) ∑ (-1)n-1 7n
n=1
Determine whether the series converges.
∞
1
47) ∑
ln 6 n
n=1
Find the slope of the polar curve at the indicated point.
π
48) r = 1 - sin θ, θ = 3
Find the area of the specified region.
49) Inside one leaf of the four-leaved rose r = 9 sin
2θ
3
4
6
Answer Key
Testname: MA2414FINALS
1)
31
5
2) 36
3) C
9
4) π
2
5)
64
π
3
6) 24
7) 16π2
8) 1200 in · lb
9) 230.1 ft · lb
10) x = 0, y = 36
11) 486w
1
12)
9x2/3
13) g(x) is a reflection of f(x) across the y-axis. It will be one-to-one.
1 - 7ln x
14)
x8
15)
1
x ln 5x
16)
2
ln 1 + 3x + C
3
17)
1
ln 2 + 9 ln x + C
9
18)
19)
1
2
1
x 1
- x + 5 x x + 5
1
- 1
θ
20) (-6θ5 e-θ6 ) cos e-θ6
1
21) e(9 sin 5x) + C
5
22) - 23)
e1/x
+ C
2
π
(ln 5θ)π-1
θ
24) (cos x)x (ln cos x - x tan x)
10
25)
2
4x + 25
26) sin-1 27)
7
5
x + 64 - 49x2 + C
8
49
7
5
1
tan-1 x + ln 4 + 49x2 + C
2
49
2
28) sin-1 (x + 3) + C
4
Answer Key
Testname: MA2414FINALS
29) y = 6 cos-1 x - 5
30) -∞
31) 18 sinh 9x cosh 9x
-10
32)
sin x
33)
1
cosh (4x - ln 3) + C
2
34)
e2x
[6 sin 6x + 2 cos 6x] + C
40
35)
1
1 5
x ln 8x - x5 + C
25
5
36) - 37)
38)
39)
csc 4t cot 4t 1
- ln csc 4t + cot 4t + C
8
8
14
15
x
4 4 + x2
1
16
40) - + C
x2 - 16
+ C
x
9 - t2
+ C
9t
41)
4
3
42)
7 2
x - 8 ln x + 1 + 8 ln x + C
2
43)
π
6
44)
π
2
45) Diverges
46) Converges; 1
2
47) converges
48) 1
81π
49)
8
5
Answer Key
Testname: MA2414FINALS
50)
6
4
2
-6
-4
-2
2
4
6
-2
-4
-6
6