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(MATH014)[2010](s)final~ywangae^_10395.pdf
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HKUST
MATH 014 Calculus II
Final Examination Name:
20th May 2009 Student I.D.:
16:30C19:30 Tutorial Section:

1.
Do not open the exam until instructed to do so.

2.
When instructed to open the exam, please check that you have 13 pages of questions in addition to the cover page.

3.
Write your name, and other information in the space provided above.

4.
Show an appropriate amount of work for each problem. If you do not show enough work, you will get only partial credit.

5.
This is a closed book and notes examination.

6.
You may write on the backside of the pages, but if you use the backside, clearly indicate that you have done so.

7.
Please turn o. all phones and pagers and remove headphones.

8.
Cheating is a serious o.ense. Students caught cheating are subject to a zero score and other penalties.


Question No. Points Out of
Q. 1 8
Q. 2 8
Q. 3 8
Q. 4 8
Q. 5 14
Q. 6 14
Q. 7 12
Q. 8 14





Q. 9
Total Points
1. ([8pts]) Considertheregionenclosedbythecurveand the x-axisasshowninthefollowing .gure.


(a) Use the trapezoidal rule with n =8 subintervals to estimate the area of the enclosed region. [3 pts]
(b) Usethediskmethod(washermethod) and midpoint rule with n =3 subintervals to estimate the volume of the solid obtained by rotating the region about the y-axis. [5 pts]
2. ([8 pts]) Use integration by parts, or otherwise, to show the following.


(a) If afunction f satis.es f(1)= f(1)=0, fis continuous on theinterval[0,1] and |f(x)|4,
. 1 . 2
show that f(x)dx . [4 pts]
0 3
1
(b) Let In = (1.x 2)ndx, for n =0,1,2,3,... . Show that In+1 = CnIn for some constant Cn
0
22n (n!)2
depending on n, and then show In = . [4 pts]
(2n +1)!
3. ([8 pts]) b is a constant in each of the following improper integrals. Determine the value of the constant b for which the corresponding improper integral converges. Show your reasoning for full credit.
3
x
(a) dx [4 pts]
1+xb
0
..
bx 1
(b) . dx [4 pts]
1 x2 +1 2x 4. ([8pts]) Matcheach of the three systems of equations with one of the four direction-.elds/vector-
dy
.elds for shown below.
dx
(A) (B)
5 5
4 4
3
3
y

y
2 2
1 1
0 012345 12 345
xx
(C) (D)
5 5
4 4
3
3
y

y
2 2
1 1
0 012345 12 34 5
xx
dx dx dx
= y.1 =5x .2.5xy = .3x + y+7
dt dt dt
Equation
dy dy dy
= sin(x .1) =3xy .3y = .2y+2
dt dt dt
Graph [6 pts]
Find alsoall constantsolutions(equilibriumsolutions) tothesystem: [2pts]
dx dy
=5x .2.5xy , =3xy .3y
dt dt
5. ([14 pts]) Atrough of 6 meters long is .lled with water and its vertical ends have the shape of the parabola region given by y = x2, for .1 x 1(in meter).

(a) Find the lateral surface area of the trough (without the two ends). [5 pts]
(Hint: consider the thinarea above a tiny arc along the parabola.)
(b) Find the hydrostatic force on each end (par