Name
Sec
MATH 152
ID
Final Exam
Sections 513 - 515
1-11
/66
Fall 2007
12
/12
P. Yasskin
13
/15
14
/10
Total
/103
Multiple Choice: (6 points each)
1. A plate is bounded by the curves y = x 2 , y = −x 2 and x = 2 measured in meters.
Find the total mass of the plate if the surface density is ρ = 3 kg/m 2 .
a. 8
3
b. 16
3
32
c.
3
d. 8
e. 16
2. A plate is bounded by the curves y = x 2 , y = −x 2 and x = 2 measured in meters.
Find the center of mass of the plate if the surface density is ρ = 3 kg/m 2 .
a.
b.
c.
d.
e.
3 ,0
4
3, 1
4 10
3, 6
4 5
3 ,0
2
3, 6
2 5
1
3. Compute
∫ x 2 e 2x dx.
2
a. x e 2x − x e 2x − 1 e 2x + C
b.
c.
d.
e.
2
x2
2
x2
2
x2
2
x2
2
e 2x −
e 2x −
e 2x −
e 2x −
2
x e 2x +
2
x e 2x −
2
x e 2x +
2
x e 2x −
4
4
1 e 2x + C
4
1 e 2x + C
2
1 e 2x + C
2
1 e 2x + C
4
x = 2t 2
y = t3
for 0 ≤ t ≤ 1 is rotated about the y-axis.
Which integral gives the area of the surface swept out?
4. The parametric curve
a. 2π
∫0 t
b. 2π
∫0 t3
16 + 9t 2 dt
c. 4π
∫0 t3
16 + 9t 2 dt
d. 2π
∫0 t4
16 + 9t 2 dt
e. 4π
∫0 t4
16 + 9t 2 dt
1
1
1
1
1
16 + 9t 2 dt
2
5. Which term appears in the partial fraction expansion of
a.
b.
c.
d.
e.
3x 2 − 4x − 20 ?
x − 2 2 x 2 + 4
−2
x − 2 2
1
x − 2 2
2
x − 2 2
−2
x − 2
1
x − 2
6. Find the solution of the differential equation x
y1 = 2.
dy
= 2y + x 3 satisfying the initial condition
dx
a. y = x 3 − 7 x 2
4
b. y = x + x 2
3
c. y = 2x 3 − x 2
9 + 1 x3
5
5x 2
5
e. y =
+ 1 x3
2
3
3x
d. y =
3
7. The region in the first quadrant bounded by the curves y = x 2 , y = 0 and x = 2 is rotated
about the y-axis. Find the volume of the solid swept out.
a. 32 π
5
b. 64 π
5
c. 8π
d. 4π
e. 2π
1, 3
by its 3 rd degree Taylor
2 2
polynomial centered at x = 1, namely T 3 x = x − 1 − 1 x − 1 2 + 1 x − 1 3 , then the
2
3
Taylor Remainder Theorem says the error |R 3 x| is less than
8. If you approximate fx = lnx on the interval
Taylor Remainder Theorem:
If T n x is the n th degree Taylor polynomial for fx centered at x = a
then there is a number c between a and x such that the remainder is
f n+1 c
R n x =
x − a n+1
n + 1!
a. 1
b. 1
2
c. 1
4
d. 1
8
e. 0
4
9. Compute lim
x0
sinx 2  − x 2
.
3
e x − 1 − x3
a. − 1
b.
c.
d.
e.
6
1
6
2
3
−1
3
1
3
10. Find the volume of the parallelepiped with edge vectors
⃗
a = −2, 2, 1,
⃗
b = 3, 2, 4
and
⃗c = 1, −2, 3
a. −26
b. 26
c.
26
d. −46
e. 46
11. If ⃗
u points Down and ⃗v points North West, in which direction does ⃗
u × ⃗v point?
a. North East
b. South
c. South West
d. South East
e. Up
5
Work Out: (Points indicated. Part credit possible.)
1
x2
12. (12 points) Compute ∫
dx
0
4 − x2
13. (15 points) A cone of radius 4 and height 6 and vertex down is filled with water up to
height 2. Find the work done to pump the water out the top. Give your answer as a multiple
of ρg.
6
14. (10 points) Start from the Maclaurin series:
a. (6 pt) Find the Maclaurin series for
1 =
1+x
∞
∑−1 n x n
n=0
1
.
1 + x 2
HINT: Differentiate both sides of the given series.
∞
b. (4 pt) Compute
∑
n=1
−1 n−1 n
.
2 n−1
7