Math 172
Exam 1 Review
Feb 2016
The problems on this page form rough approximations of an exam.
1. Integrate. (Sunday)
Z 3
√
(a)
t2 1 + t dt
Z
(b)
0
2. Integrate. (Monday)
Z eπ
sin (ln t)
(a)
dt
t
1
Z
(b)
Z
dy
y2 + 4
(c)
x arctan x dx
Z
π/2
sin3 2θ dθ
(d)
0
Z
dy
p
2 − y2
(c)
x ln x dx
Z
(d)
π/4
sec4 θ dθ
0
3. Consider the area between the graphs of the functions y = sin x and y = cos x for x ∈ [0, π/2].
Carefully sketch the graphs and find the area.
4. Find the volume of the solid with base bounded by y = |2x| and y = 2 with cross sections
perpendicular to the y-axis that are squares.
5. Consider the region between the graph of y = ex and the x-axis for x ∈ [0, 1]. Carefully sketch
the region. Find the volume of the solid generated by rotating the region about the y-axis.
6. Consider the region bounded by the graphs of y = x2 and y = x + 2. Carefully sketch the region.
Find the volume of the solid generated by rotating the region about the line y = −1.
7. Calculate the work (in joules) required to pump all of the mystery fluid of density ρ out of a full
tank on the planet Zanabar with gravitational constant g. The tank is in the shape of the graph
of y = x4 for x ∈ [0, 2] rotated about the y-axis. The mystery fluid is pumped out a spout of
length 1 m from the top of the tank.
8. Integrate. (Sunday)
Z
t dt
√
(a)
4 − t2
9. Integrate. (Monday)
Z
√
(a)
t 2 − t dt
Z
1/4
(b)
1/8
Z
(b)
0
1/3
dy
p
1 − 16y 2
dy
2
9y + 1
Z
π/4
x sec2 x dx
(c)
Z
sin4 θ dθ
Z
sec4 θ tan θ dθ
(d)
0
Z
(c)
x2 sin 2x dx
(d)
10. Consider the area between the graphs of the functions y = x2 and y = x for x ∈ [0, 2]. Carefully
sketch the graphs and find the area.
11. Find the volume of the solid with base bounded by y =
sections perpendicular to the x-axis that are semicircles.
1
x
and x-axis for x ∈ [1, 3] with cross
12. Consider the region bounded by the graphs of y = x2 , y = 12 − 4x, and the x-axis. Express the
volume of the solid generated by rotating the region about the x-axis as an integral using both
the disk/washer/slicing method and the shell method. Do not evaluate either integral.
13. Consider the region bounded by the graphs of x = y 2 and x = y + 2. Carefully sketch the region.
Find the volume of the solid generated by rotating the region about the line y = 2.
14. Calculate the work (in joules) required to pump all of the mystery fluid of density ρ out of a full
tank on the planet Zanabar with gravitational constant g. The tank is a horizontal cylinder of
radius r and length l. The mystery fluid is pumped out of the top of the tank.
Additional review problems.
1. Integrate.
Z e
ln x
(a)
dx
x
1
Z
(e)
e2x sin 3x dx
Z
(b)
0
3
Z 1
Z π/2
1+x
1 + 2x
√
dx
(c)
dx
(d)
x cos x dx
9 + x2
1 − x2
0
0
Z
Z
(f)
sin x cos x sin 2x dx
(g)
sec3 x tan3 x dx
2. Consider the area between the graphs of the functions y = tan x and y = sec x for x ∈ [0, π/4].
Carefully sketch the graphs and find the area.
3. Find the volume of the solid with base bounded by y = tan x and x-axis for x ∈ [0, π/4] with cross
sections perpendicular to the x-axis that are squares.
4. Consider the region bounded by the graph of y = sin (x/2) and the x-axis for x ∈ [0, 2π].
(a)
(b)
(c)
(d)
Find
Find
Find
Find
the
the
the
the
volume
volume
volume
volume
of
of
of
of
the
the
the
the
solid
solid
solid
solid
generated
generated
generated
generated
by
by
by
by
rotating
rotating
rotating
rotating
the
the
the
the
region about the x-axis.
region about the y-axis.
region about the line y = 1.
region about the line x = −1.
√
5. The region in the first quadrant bounded by the graphs of y = x and y = 2 − x, the shaded
region in Figure 1 below, is revolved around the x-axis. Express the volume of the resulting solid
as an integral, or integrals, using the Disk Method and the Shell Method. Evaluate the easier of
the two.
6. The region in the first quadrant bounded by the graphs of y = 5 − x2 and y = 6/x − 2, the shaded
region in Figure 2 below, is revolved around the line x = −1. Express the volume of the resulting
solid as an integral using the Disk Method and the Shell Method. Evaluate the easier of the two.
4
1
3
2
1
1
2
-1
Figure 1:
0
1
2
Figure 2:
7. Show the volume of a sphere of radius R is V = 43 πR3 .
8. Calculate the work (in joules) to build a pyramid with square base of side length s and height h
out of a material of density ρ with a gravitational constant g.
9. Let’s build a snowman!
(a) Calculate the work (in joules) to buld a sphere of radius r with center at height h out of
snow of density ρ with a gravitational constant g.
(b) Using part (a) find the work to build a snowman if the base is a sphere of radius 1/2 m, the
middle sphere has radius 1/3 m, and the top sphere has radius 1/4 m.
(c) Adding a top hat, coal for eyes, and a carrot nose is up to you.