Math 172
Exam 2 Review
March 2016
Standard problems.
1. Integrate the following trigonometric integrals.
Z
Z π
3
(c)
sin2 πx dx
(a)
sin 5x dx
0
Z
Z
(b)
sin9 x cos3 x dx
(d)
tan2 x sec2 x dx
Z
tan6 x sec4 x dx
Z
sec3 2x tan3 2x dx
(e)
(f)
2. Integrate the following using an appropriate trigonometric substitution.
Z p
Z
Z
x3
x2
2
(a)
4 − x dx
√
√
dx
dx
(e)
(c)
x2 − 4
1 − x2
√
Z 2
Z
Z
x2 − 3
9
dx
(b) √
dx
√
(f)
dx
(d)
2
x
(9
+
x2 )2
4x + 1
3
3. Integrate the following rational functions using the method of partial fraction decomposition.
Z
Z 3
Z
3−x
x − 4x2 + 10x
3x2 + x + 9
dx
(a)
dx
(e)
dx
(c)
1 − x2
x3 + 9x
x2 − 4x + 8
Z 0
Z
Z
3
4x2 + 5x + 3
sin x
(b)
, dx
(d)
dx
(f)
dx
2
2
x(x + 1)
cos x − cos2 x
−1 x + x − 2
4. Evaluate the following improper integrals, or show they diverge.
Z ∞
Z 0
Z 1
2
x
(c)
dx
(a)
xe dx
(e)
ln x dx
3x + 5
0
−∞
0
Z 2
Z ∞
Z ∞
dx
5
arctan x
√
(f)
(b)
dx
(d)
dx
5
2
2
x−2
x −x−6
x +1
−5
4
0
5. Use the Comparison Theorem to show the following integrals converge or diverge.
Z ∞
Z π/2
Z ∞ −x2
dx
cos x
e
(a)
√ dx
(c)
(e)
dx
4+x+7
x
1
x
x2 + 1
0
2
Z ∞
Z 2
Z 2 x2
dx
2
e
√ dx
√
(b)
(f)
dx
(d)
2
2
x− x
2
0 x + x
0 x
6. Find C so that the following are probability density functions on the given interval. Find the
specified probability.
(a) p(x) = Cx2 on [0, 3]; P (1 ≤ X ≤ 2)
C
on [0, ∞); P (X ≥ 1)
(b) p(x) = 2
x +1
C
(c) p(x) = √
on [0, 2); P (0 ≤ X ≤ 1)
4 − x2
7. Calculate the arc length of the curve over the given interval.
(a) y = 1 + 3x,
[0, 2]
1
(b) y = (2 + x2 )3/2 ,
3
(c) y = 1 + x2 ,
[0, 2] (d) y =
x4
1
+ ,
32 x2
[0, 2]
[1, 2]
(e) y = ln cos x,
(f) y =
x5
1
+
,
20 3x3
[0, π/3]
[1, 2]
8. Calculate the surface area generated by rotating the given curve about the x-axis over the given
interval.
√
√
(a) y = x3 , [0, 1]
(b) y = 4 − x2 , [−1, 1]
(c) y = x, [0, 1]
9. The end of trough has the shape of a trapezoid with width 2 m at the bottom, width 4 m at the
top, and height 2 m, see Figure 1 below. If the trough is filled to the top with orange juice with
kg
density ρ = 1150 m
3 , find the fluid force on the trapezoidal end of the trough.
10. Calculate the fluid force of a plate in the shape of the region shown in Figure 2 below. The surface
of the mystery fluid of density ρ is at y = 1.
11. Calculate the fluid force on a semicircular plate of radius 1 m centered at the origin and oriented
as shown in Figure 3 below. The surface of the mystery fluid of density ρ is at y = 2.
4
fluid level
2
fluid level
1
1
2
0
0
6
1
e
y=ln(x)
0
-1
2
Figure 1: A trough.
Figure 2: A plate.
0
Figure 3: A plate.
The following information is given on the exam.
Some trigonometric identities which may or may not be needed include:
sin A cos B =
1
cos2 x =
(1 + cos 2x)
2
1
sin2 x =
(1 − cos 2x)
2
sin 2x = 2 sin x cos x
cos A cos B =
sin A sin B =
1
(sin(A + B) + sin(A − B))
2
1
(cos(A + B) + cos(A − B))
2
1
(cos(A − B) − cos(A + B))
2
Some integrals which may or may not be needed include:
Z
Z
du
2
a + u2
√
du
− u2
a2
u
1
arctan
+c
a
a
u
= arcsin
+c
a
=
Z
sec u du = ln |sec u + tan u| + c
Z
sec3 u du =
1
1
sec u tan u + ln |sec u + tan u| + c
2
2
Z
csc u du = ln |csc u − cot u| + c
1
supplemental problems.
Although many of these problems are similar to problems that are reasonable for the exam, a
few of these problems will stretch your brain/patience in ways that the exam may not; master
the earlier examples first.
12. Integrate. For improper integrals, either show it diverges or evaluate it.
Z
Z 4
Z r
dx
1+x
(a)
cos5 3x sin2 3x dx
√
(o)
(h)
dx
3
x
1−x
−1
Z 3
Z
Z
2
x + x − 9x − 19
3x2 + 11x + 16
4
(b)
dx
(i)
cos
2x
dx
(p)
dx
x2 − x − 6
(x + 1)(x2 + 6x + 13)
Z
Z
Z
p
5x2 − 3x + 5
ex
(c)
ex 1 − e2x dx
dx
(j)
(q)
dx
3
x + 5x
e2x − ex
Z
Z √
Z
x
e2x
9
3
(d)
sec x tan x dx
(k)
dx
(r)
dx
x−4
e2x − ex
Z
Z
Z 1p
p
2
2
2
(l)
sin
x
cos
x
dx
(e)
sin x 1 + cos x dx
4 + x2 dx
(s)
0
Z
Z
Z
2x
6x2 − 8x + 56
dx
(m)
(t)
sec x tan5 x dx
(f)
dx
2 − 6x + 13)2
4
(x
x − 16
Z 4
Z
Z
dx
2x2 + 11x + 11
6
(n)
(u)
dx
(g)
sec 4x dx
2
(x + 1)(x + 2)(x + 3)
−1 x
13. Use the Comparison Theorem to show the following integrals converge.
Z ∞
Z ∞
Z ∞
dx
2
|sin x|
√
(a)
(c)
e−x dx
(b)
dx
2
2
x + x
x +1
0
−∞
π
14. Calculate the fluid force on one side of a plate in the shape of the region bounded by the graph
of y = sin x and the x-axis for x ∈ [0, π], see the figure below. The surface of the mystery fluid of
density ρ is at y = 2.
y
2
fluid level
1
plate
0
π/2
π x
15. Calculate the fluid force on the infinite plate ‘bounded’ by the curve y = ln x, the x-axis, and the
y-axis in the fourth quadrant. The surface of the mystery fluid of density ρ is at y = 0.
16. Calculate the surface area of the curve y = sin x on the interval [0, π] rotated about the x-axis.
17. Take a break, seriously.