Name
1-10
/50
11
/10
Fall 1999
12
/10
P. Yasskin
13
/10
14
/10
15
/10
ID
MATH 172
FINAL EXAM
Section 502
Multiple Choice: (5 points each)
1.
Find the area below
y
Ÿ 3 sin 2x
0 t x t =.
2
above the x-axis for
3
2.5
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
x
1
1.2
1.4
a. 1
b.
c.
d.
e.
2
3
2
=
2
3=
2
3
Ÿ is rotated
0txt =
2
about the y-axis. (See the figure in problem 1.) Which formula will give the volume of
the solid of revolution?
2. The region below
a. A
b. A
c. A
d. A
e. A
;
;
;
;
;
=/2
0
=/2
0
=/2
0
=/2
0
=/2
0
y
3 sin 2x
above the x-axis for
Ÿ 3x sinŸ2x dx
6=x sinŸ2x dx
9= sin Ÿ2x dx
18= sin Ÿ2x dx
x 2 sin 2x dx
2
2
1
3. A 1 m bar has linear mass density
>
kg
1
1 x2 m
where x is measured from one
>
kg
1
1 x2 m
where x is measured from one
end. Find the total mass.
a. M
b. M
c. M
d. M
e. M
= kg
4
= kg
2
1 kg
2
45 kg
90 kg
4. A 1 m bar has linear mass density
end. Find the center of mass.
a. x
b. x
c. x
d. x
e. x
ln 2 m
90
ln 2 m
2
2 ln 2 m
=
ln 2 m
2=
1 m
2
;
5. Compute
=/2
"=/2
sin 6 2 cos 2 d2
a. " 2
7
b. " 1
7
c. 0
d. 1
7
e. 2
7
for
is rotated about the x-axis. Which formula
y x3
0txt3
will give the area of the surface of revolution?
6. The curve
a. A
b. A
c. A
d. A
e. A
;
;
;
;
;
3
0
3
0
3
0
3
0
3
0
2=x 1 9x 4 dx
2=x 3 1 9x 4 dx
2=x 3 dx
Ÿ 2=x 3x 2 dx
=x 1 9x 4 dx
2
;
7. Compute
2
0
2x dx
4 " x2
a. ".
b. " ln 4
c. =
4
d. ln 4
e. .
8. If it requires 24 J of work to stretch a spring from rest to 4 m, how much work will it take
to stretch it from 2 m to 6 m?
a.
b.
c.
d.
e.
6J
12 J
24 J
48 J
96 J
9. Which term is incorrect in the following partial fraction expansion?
Ÿ
x 3 " 2x 3
x " 2 2 x " 3 x2
Ÿ
Ÿ
4
A
x"2
a.
Ÿx " 2 Bx C
2
D
x"3
b.
c.
Ex F
x2 4
d.
e. They are all correct.
!
.
10. Find the radius of convergence of the series
n1
Ÿ
n2 x " 2 n.
3n
a. 0
b. 1
3
c. 3
d. 9
e. .
3
11. (10 points) Compute
;
=/2
0
Ÿ 3x cos 2x dx
12. (10 points) Find the length of the parametric curve given by
1 t4
for 0 t t t 2.
4
HINT: Factor the quantity in the square root.
z
x
t2,
y
2 t3,
3
4
13. (10 points) Find the volume of the solid whose base is the semi-circle
x2 y2 9
squares.
for y u 0
and whose crosssections perpendicular to the x-axis are
14. (10 points) Solve the differential equation
condition
Ÿ y3
dy
dx
1 x2
y2
y2x2
with the initial
0.
5
15. (10 points) Given the series
ex
a. (5 pts) compute the series for
b. (5 pts) and use it to compute
1 x 1 x2
2
e 2x
1 x3
6
C,
2x
lim e " 12 " 2x .
xv0
x
(2 pts only for l’Hospital’s Rule.)
6