MATH 152 Exam 1 Review Activity (Activity 5) (Sections 6.4-8.1)
Directions: Put both your name and your partners name on the answersheet. Today, you will only work problems
1-10. Put the letter of the correct response on the answersheet. Calculators are not allowed and you may use your notes
and textbook. Failure to follow these instructions will result in a 1 point deduction.
PART I: Multiple Choice
1. Find the average value of f (x) = x sin(x2 ) from x = 0 to x =
√
π.
(a) 1
(b) −1
1
(c) − √
π
1
(d) √
π
(e) 0
2.
Z
2
x3 ln x dx =
1
15
16
3
4 ln 2 −
4
4 ln 2
3
4 ln 2 +
4
15
4 ln 2 +
16
(a) 4 ln 2 −
(b)
(c)
(d)
(e)
3. A 50 foot rope that weighs 25 pounds hangs from the top of a tall building. How much work is required to pull 10
feet of the rope to the top?
(a) 25 foot pounds
(b) 900 foot pounds
(c) 100 foot pounds
(d) 225 foot pounds
(e) 120 foot pounds
1
4. A solid S has a base which is a circular disk of radius r. Parallel cross sections perpendicular to the base are squares.
Find the volume of the solid.
a.) 163r3
b.) 83r3
c.) 43r3
d.) 23r3
e.) 32r3
5. The region bounded by y = x2 and y = 2x is revolved about the y-axis. Find the volume.
(a)
(b)
(c)
(d)
(e)
4π
3
8π
15
64π
15
2π
3
8π
3
6. Find the area of the region bounded by y = x2 and y = 8 − x2 .
(a)
(b)
(c)
(d)
(e)
32
3
64
3
22
3
44
3
128
3
7. Find the positive value of b so that the average value of f (x) = 3x2 − 2x over the interval [0, b] is equal to 2.
(a) b = 1
(b) b = 2
4
(c) b =
3
3
(d) b =
4
(e) Not enough information.
1
8. Find the volume of the solid obtained by revolving the region bounded by y = , y = 0, x = 1 and x = 5 about
x
the y-axis.
(a) 23π
(b) 8π
2
(c) 2π ln 5
4π
(d)
5
2π
(e)
5
9.
Z
π/4
(sec2 x)etan x dx =
0
(a) e
√
2/2
(b) e
1/2
(c) e
√
2
−1
−1
−1
(d) 1 − e
(e) e − 1
10. Which integral is the volume of the solid generated by revolving about the y-axis the region enclosed by the curves
y = 1 − x2 and y = (x − 1)2 ?
R1
R1
(a) π 0 [(1 − x2 )2 − (x − 1)4 ]dx (b) 2π 0 x(2x − 2x2 )dx
(c)
R1
0
(e) 2π
[1 − x2 − (x − 1)2 ]dx
R1 √
√
y + 1)dy
0 y( 1 − y −
(d)
R1 √
√
( 1 − y − y + 1)dy
0
11. The region bounded by y = x2 and y = 4 is revolved around the x-axis. Find the volume of the solid obtained.
(a)
(b)
(c)
(d)
(e)
256π
5
704π
5
128π
5
88π
3
128π
3
3
12.
Z
1
x sin(πx) dx =
0
1
π
(b) π
(a)
1
π
(d) −π
(c) −
(e) 0
PART II WORK OUT This portion is not to be turned in today. It’s just for practice.
13. Find the volume of the solid obtained by rotating the region bounded by y = x2 + 1 and y = 2 about the line y = 3.
14. A spring has a natural length of 2 meters. If the work required to stretch the spring from 3 meters to 4 meters is 8
Joules, find the work done in stretching the spring from 4 meters to 5 meters.
15. A triangluar trough is 8 meters long. The end of the trough has the shape of an isosceles triangle 2 meters tall and
1
1 meter wide (see figure). Find the work needed to pump all the water out of a spout that is m high. Note: The
2
weight density of water is ρg = 9800 Newtons per cubic meter.
1m
2m
16. Find the volume of the solid described here: The base of the solid is the region bounded by y = x2 and y = 4. Cross
sections perpendicular to the y-axis are equilateral triangles.
17. Consider the region bounded by y = sin x, y = cos x, x = 0 and x = π.
(i) Shade the bounded region. Be sure to clearly label all pertinent points.
(ii) Find the area of this bounded region.
4