Name
ID
MATH 172
EXAM 1
Spring 1999
Section 504
Solutions
P. Yasskin
Multiple Choice: (6 points each)
1. Evaluate
a. 6 9=
3
; 2 9 " x 2 dx
1-10
/60
11
/10
12
/10
13
/10
14
/10
by interpretating it as an area.
0
correctchoice
4
b. 3 9=
c. 2 9=
2
d. 2 9=
3
e. 6 9=
; 2 9 " x 2 dx rectangle quarter circle
5
0
3 2 1 =Ÿ3 2 6 9=
4
4
4
3
2
1
0
2. Compute:
a. 47
;
2
1
1
2
3
x 3 x 13
x
dx
8
b. 45
8
c. 351
64
415
d.
64
417
e.
64
;
2
1
correctchoice
x 3 x 13
x
dx x4 x2 " 1
4
2
2x 2
2
1
42" 1
8
"
1 1 " 1
2
4
2
45
8
1
;
3. Compute:
=/4
0
cos 3 2 sin 2 d2
a. " 3
16
b. " 1
16
1
c.
16
d. 3
16
e. 3
correctchoice
u cos 2
;
=/4
0
du " sin 2 d2
4
" 1
1
2
4
=/4
4
4
" cos 2
u 3 du " u
4
4
2 0
1 " 1 1 3
16
4
4
16
cos 3 2 sin 2 d2 " ;
=/4
0
2
d ; x e t 2 dt
dx x
4. Compute:
a. e x " e x
4
2
b. e x 2x " e x
4
2
correctchoice
c. e 4x " e
2
2
d. e 4x " e x
2
2
e. e 4x 2x " e x
x4
3
x2
x2
Let FŸt be an antiderivative of e t . Then F U Ÿt e t . So ; e t dt FŸx 2 " FŸx .
2
2
2
2
x
x
2
4
2
Thus by chain rule: d ; e t dt F U Ÿx 2 2x " F U Ÿx e x 2x " e x
dx x
5. Compute:
a.
b.
c.
d.
e.
;
x 2 1 dx
2
3
Ÿx 3x "3 C
x 3x
"1 C
x 3 3x
"1
C
3x 3 9x
1
C
x 3 3x
3
C
x 3 3x
3
u x 3 3x
;
correctchoice
du Ÿ3x 2 3 dx 3Ÿx 2 1 dx
x 2 1 dx 1 ; 1 du "1 C "1
C
2
2
3
3
3
3u
3x
9x
u
3x
Ÿx
1 du Ÿx 2 1 dx
3
2
e
; x 4 lnx dx
1
4 e5 1
correctchoice
25
25
4 e5 " 1
25
25
4 Ÿe 5 " 1 25
4 Ÿe 5 1 25
6. Compute:
a.
b.
c.
d.
u lnx
dv x 4 dx
5
du 1x dx v x
5
e. 1 Ÿe 5 1 5
e
5
4
; x 4 lnx dx x5 lnx " ; x5 dx
1
5
5
e lne " e
" 1 ln 1 " 1
5
5
25
25
1
e
x 5 lnx " x 5
5
25 1
1 " 1 e5 1 4 e5 1
5
25
25
25
25
y 12 " x 2
7. Find the area between the curves
a. 2
b. 4
c. 12
d. 24
e. 32
e
and
y y 2x 2 .
12 " x 2 2x 2
correctchoice
12 3x 2
10
x o2
5
-4
A;
2
"2
Ÿ12
0
-2
" x 2 " Ÿ2x 2 dx ;
2
"2
Ÿ12
2x
4
" 3x 2 dx ¡12x " x 3 ¢ "2 2 32
3
1
8.
y sinŸx
The region below
0txt
for
=
2
above the x-axis
is rotated about the y-axis.
Find the volume of the solid swept out.
-2
-1
0
1
x
2
a. =
2
b. =
c. 2=
correctchoice
d. 3=
e. 4=
Thin cylinders with radius r x and height h sinŸx 2 .
V;
=
0
2=x sinŸx 2 dx " = cosŸx 2 =
0
"= cos = = cos 0 2=
9. The force needed to stretch a superspring
xm
beyond its natural length is
how much
If it takes
to stretch the superspring by
F
2 m,
32 N
to
work is done in stretching it from
1m
2m?
a. 3
b. 7
c. 15
correctchoice
d. 31
e. 63
kx 3 .
32 k 2 3 8k
F kx 3
2
2
1
1
W ; Fdx ; 4x 3 dx x 4
k4
15
10. The mass density of a 9 m rod is > Ÿ1
average density of the rod.
kg
a. .495 m
kg
b. .99 m
kg
c. .4995 m
F 4x 3
kg
18
x 3 m
for 0 t x t 9. Find the
correctchoice
kg
d. .999 m
kg
e. .49995 m
9
18
> ave 1 ;
dx 2
9 0 Ÿ1 x 3
Ÿ1
x "2
"2
9
0
"1
2
Ÿ1 x 9
0
"1 " "1 .99
100
1
4
11. (10 points)
1
; arctan2x dx
0 1x
Compute:
1
; arctan2x dx
0 1x
;
=/4
u arctan x
u du
0
2
u
2
=/4
0
2
=
1 dx
1 x2
x 1 @ u arctan 1 =
4
x 0 @ u arctan 0 0
du 32
OR
;
1
2
u du u
2
x0
12. (10 points)
1
x0
Ÿarctan x 2
2
1
Ÿarctan 1 0
2
2
"
Ÿarctan 0 2
2
2
=
32
; sec 3 2 tan 3 2 d2
Compute:
; sec 3 2 tan 3 2 d2 ; sec 2 2 tan 2 2 sec 2 tan 2 d2
du sec 2 tan 2 d2
; u 2 Ÿu 2 " 1 du
u sec 2
;Ÿu 4 " u 2 du
tan 2 2 sec 2 2 " 1 u 2 " 1
5
3
5
3
u " u C sec 2 " sec 2 C
5
13.
5
3
3
(10 points)
A hemispherical bowl of radius
is filled to a depth of
2 ft
1 ft.
Find the volume of the water.
HINT: Slice it horizontally.
Put the zero of the vertical y-axis at the center of the sphere and the positive direction
down. Then the water goes from y 1 to y 2. A slice perpendicular to the y-axis at
height y is a disk of radius r 4 " y 2 .
2
2
2
2
y3
5=
dy ; =Ÿ4 " y 2 dy = 4y "
V ; =r 2 dy ; = 4 " y 2
3 1
3
1
1
14. (10 points)
How much work is done in pumping the water out the top of the bowl
shown in problem 13? Leave the density as > and the acceleration of gravity as g.
2
The volume of the slice at height y is dV = 4 " y 2
dy =Ÿ4 " y 2 dy. So its weight
is dF >g dV >g =Ÿ4 " y 2 dy. This slice must be lifted a distance D y. So the work
2
2
2
y4
done is W ; D dF ; y >g =Ÿ4 " y 2 dy >g = ; Ÿ4y " y 3 dy >g = 2y 2 "
4 1
1
1
9
>
g
=
>g =¡8 " 4 ¢ " >g = 2 " 1 4
4
5