Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam1
Fall 2000
Test Form A
Solutions
1-10
/50
11
/10
12
/10
Part II is work out. Show all your work. Partial credit will be given.
13
/10
You may not use a calculator.
14
/10
15
/10
Part I is multiple choice. There is no partial credit.
TOTAL
Formulas:
sinŸA B sin A cos B cos A sin B
cosŸA B cos A cos B " sin A sin B
sin 2 A 1 " cos 2A
2
1
cos
2A
2
cos A 2
sin A sin B 1 cosŸA " B " 1 cosŸA B 2
2
1
sin A cos B sinŸA " B 1 sinŸA B 2
2
1
cos A cos B cosŸA " B 1 cosŸA B 2
2
; sec 2 d2 ln|sec 2 tan 2 | C
; csc 2 d2 ln|csc 2 " cot 2 | C
; lnx dx x lnx " x C
1
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator.
1. What is the average value of the function fŸx x 3 on the interval ¡"1, 2 ¢.
a. 5
b.
c.
d.
e.
4
15
4
4
4
3
20
3
correctchoice
2
f ave 1 ; x 3 dx 1
3
3 "1
x4
4
2
"1
1 ¡16 " 1 ¢ 15 5
12
2. Using a trigonometric substitution, the integral
a. ; sec 2 d2
12
4
;
dx
9 x2
becomes:
correctchoice
d2
3 sec 2
c. ; d2
sec 2
d. ; 3 tan 2 d2
b. ;
e. ; d2
tan 2
9 x2 x 3 tan 2
dx 3 sec 2 2 d2
2
dx
; 3 sec 2 d2 ; sec 2 d2
;
3
sec
2
2
9x
3.
9 9 tan 2 2 3 sec 2
The region bounded by the curves
y sinŸx 2 , y 0, x 0 and x =
is rotated about the y-axis. Find the volume.
a. 1
b. 2
c. =
d. 2=
correctchoice
e. 4=
2
x-integral, cylinders, radius r x,
height h sinŸx 2 V ; 2=r h dx ;
=
0
2=x sinŸx 2 dx
du 2x dx.
Let u x 2
=
V ; = sinŸu du = " cos u
0
=
0
=Ÿ" "1 " "1 2=
4.
A tank has the shape of a half cylinder which
is 5 m long and 2 m in radius laying on its
side. The tank is full of water. Which integral
gives the work done to pump the water out of
a spout which is 1 m above the tank.
The density of water is > 1000 kg/m 3 .
The acceleration of gravity is g 9.8 m/sec 2 .
Measure y down from the axis of the cylinder.
2
a. 9800 ; Ÿy 1 10 4 " y 2 dy
"2
2
b. 9800 ; Ÿ1 " y 5 4 " y 2 dy
"2
2
c. 9800 ; Ÿ1 " y 10Ÿ2 " y dy
0
2
d. 9800 ; Ÿy 1 5Ÿ2 " y dy
0
2
e. 9800 ; Ÿy 1 10 4 " y 2 dy
correctchoice
0
A slice of the water at depth y and thickness y must be lifted a distance D y 1. Its length
is 5 and its width is 2 4 " y 2 . So its volume is V 10 4 " y 2 y and its weight is F >g V.
So the work to lift the slice is W D F. We add up the work for the slices between y 0 and
y 2. So the total work is
2
W 9800 ; Ÿy 1 10 4 " y 2 dy
0
1
5. Compute ; Ÿx " 2 e x dx.
a. 1 " 2e
b. 1
c. 3 " 2e
0
correctchoice
d. "2e
e. 3
1
; Ÿx " 2 e x dx 0
Ÿx
1
1
0
0
" 2 e x " ; e x dx
Ÿx
" 2 ex " ex
1
0
¡Ÿ1
u x"2
dv e x dx
du dx
v ex
" 2 e " e ¢ " ¡Ÿ"2 " 1 ¢ 3 " 2e
3
6. Evaluate ; cos 3 x dx.
4
a. sin x C
4
2
4
b. sin x " sin x C
2
4
4
cos
x
C
c.
4
3
d. sin x " sin x C
3
4
e. " cos x C
4
u sin x
correctchoice
du cos x dx
cos 2 x 1 " sin 2 x 1 " u 2
3
3
; cos 3 x dx ; cos 2 x cos x dx ;Ÿ1 " u 2 du u " u C sin x " sin x C
3
3
7. Which of these integrals represents the area between the curves y sin x and y cos x
from x 0 to x =.
a. ;
b. ;
c. ;
=/4
0
=/4
0
=/4
0
Ÿcos x
" sin x dx ;
Ÿcos x
" sin x dx ;
Ÿsin x
" cos x dx ;
=
3=/4
=/4
=
=/4
=
=/4
Ÿsin x
" cos x dx ;
Ÿsin x
" cos x dx
Ÿcos x
" sin x dx
=
3=/4
Ÿcos x
" sin x dx
correctchoice
d. ; Ÿcos x " sin x dx
0
=
e. ; Ÿsin x " cos x dx
0
1
On 0 x = , cos x sin x.
4
=
x =, cos x sin x.
On
4
0
1
2
3
A;
=/4
0
Ÿcos x
" sin x dx ;
=
=/4
Ÿsin x
" cos x dx
-1
4
8. Evaluate ; x 9 x 2 dx.
0
a. 147
b.
c.
d.
e.
2
98
3
6
392
3
8
3
correctchoice
1 du x dx
2
4
25
3/2
1
2
1/2
; x 9 x dx ; u du 1 2u
2 9
2 3
0
u 9 x2
du 2x dx
25
9
1 Ÿ125 " 27 98
3
3
4
9. The base of a solid is the circle x 2 y 2 9. The cross sections perpendicular to the x-axis
are squares. Find the volume.
a. 9=
b. 36
c. 72
d. 81=
correctchoice
e. 144
x-integral
2
Side of square s 2y 2 9 " x 2
Area of square AŸx s 2 4Ÿ9 " x 2 -2
0
-2
2
Volume V ; AŸx dx ;
3
4 9x " x
3
3
"3
b.
c.
d.
e.
C
2
x
x
Ÿx " 1 A B C
x
2x 2
x3
C
A B correctchoice
2
x
x"1
Ÿx " 1 A B
2
x
Ÿx " 1 A B C D
2
x
x"1
x2
Ÿx " 1 "3
4Ÿ9 " x 2 dx
4¡18 ¢ " 4¡"18 ¢ 144
10. What is the form of the partial fraction decomposition of
a. A B2 3
x2 3 ?
x 3 " 2x 2 x
2
C
x2 3
x 32 A B 2
2
x
x"1
x " 2x x
xŸx " 1 Ÿx " 1 3
5
Part II: Work Out (10 points each)
Show all your work. Partial credit will be given.
You may not use a calculator.
11. Compute ; x 3 lnx dx.
u ln x
du 1x dx
dv x 3 dx
4
v x
4
4
4
4
x4 C
; x 3 lnx dx x4 lnx " ; x4 1 dx x4 lnx " 16
x
12. Find the area between the curves x y 2 " 1 and y x " 5.
a. (4 pts) Graph the curves.
4
3
2
1
0
-2
2
4
6
8
10
-1
-2
-3
b. (4 pts) Set up the integral(s) for the area.
y2 " 1 y 5
A;
3
"2
ŸŸy
y2 " y " 6 0
y "2, 3
5 " Ÿy 2 " 1 dy
c. (2 pts) Compute the area.
A;
3
"2
Ÿ" y
13. Compute ;
u sec x
;
=/3
0
=/3
2
3
2
y 6 dy " y y 6y
3
2
3
"2
"9 9 18 "
2
8 2 " 12
3
125
8 "2 "
3
1 "1
3
6
tan 3 x sec x dx.
0
du sec x tan x dx
tan 3 x sec x dx ;
=/3
0
tan 2 x sec 2 x " 1 u 2 " 1
2
tan 2 x sec x tan x dx ; Ÿu 2 " 1 du 1
u3 " u
3
2
1
4
3
6
14. Evaluate ;
2 /2
x2
dx.
1 " x2
0
1 " x 2 1 " sin 2 2 cos 2
dx cos 2 d2
2 /2
=/4
=/4
=/4
2
x2
dx ; sin 2 cos 2 d2 ; sin 2 2 d2 ; 1 " cos 22 d2
;
cos 2
2
0
0
0
0
1 " x2
=
sin
=/4
2
1 = "
1 2 " sin 22
"0 = " 1
2
2
8
4
2 4
2
0
x sin 2
x " 1 , x 3, x 6 and y 0.
15. Consider the region in the plane bounded by the curves
a. (3 pts) Graph the region.
2
1
0
1
2
3
4
5
6
7
b. (1 pt) The region is rotated about the x-axis. To find the volume, you will use
an x-integral
a y-integral
(Circle one.)
with
disks
Correct:
washers
x-integral with disks
cylindrical shells
OR
(Circle one.)
y-integral with cylindrical shells
c. (4 pts) Set up the integral(s) for the volume.
x-integral:
y-integral:
6
6
3
3
V ; =R 2 dx ; =
V;
2
0
x"1
5
2=rh dy ;
2
2
dx
2=rh dy ;
2
0
2=yŸ6 " 3 dy ;
5
2
2=yŸ6 " Ÿ1 y 2 dy
d. (2 pts) Compute the volume.
x-integral:
y-integral:
6
2
V = ; Ÿx " 1 dx = x " x
2
3
V;
2
0
6=y dy ;
5
2
6
3
= 36 " 6 " = 9 " 3 21 =
2
2=yŸ5 " y 2 dy 3=y 2
2
2
2
0
4
5
2
2
5=y 2 " = y
6= 25= " 25 = " ¡10= " 2= ¢ 21 =
2
2
7