Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam1
Fall 2000
Test Form A
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
cos 2 A 1 cos 2A
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
4
b. 15
4
c. 4
d. 4
3
e. 20
3
2. Using a trigonometric substitution, the integral
;
dx
9 x2
becomes:
a. ; sec 2 d2
d2
3 sec 2
c. ; d2
sec 2
d. ; 3 tan 2 d2
b. ;
e. ; d2
tan 2
3.
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.
b.
c.
d.
e.
1
2
=
2=
4=
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
0
1
5. Compute ; Ÿx " 2 e x dx.
a.
b.
c.
d.
e.
1 " 2e
1
3 " 2e
"2e
3
0
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
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
d. ; Ÿcos x " sin x dx
0
=
e. ; Ÿsin x " cos x dx
0
4
8. Evaluate ; x 9 x 2 dx.
0
a. 147
b.
c.
d.
e.
2
98
3
6
392
3
8
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=
e. 144
10. What is the form of the partial fraction decomposition of
a. A B2 b.
c.
d.
e.
C
2
x
x
Ÿx " 1 A B C
x
2x 2
x3
A B C
2
x
x"1
Ÿx " 1 A B
2
x
Ÿx " 1 D
A B C 2
x
x"1
x2
Ÿx " 1 x2 3 ?
x " 2x 2 x
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.
12. Find the area between the curves x y 2 " 1 and y x " 5.
a. (4 pts) Graph the curves.
b. (4 pts) Set up the integral(s) for the area.
c. (2 pts) Compute the area.
6
13. Compute ;
14. Evaluate ;
=/3
tan 3 x sec x dx.
0
2 /2
0
x2
1 " x2
dx.
7
15. Consider the region in the plane bounded by the curves
a. (3 pts) Graph the region.
x " 1 , x 3, x 6 and y 0.
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
washers
cylindrical shells
(Circle one.)
c. (4 pts) Set up the integral(s) for the volume.
d. (2 pts) Compute the volume.
8