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(MATH014)[2011](s)final~=yqp24^_95377.pdf
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HKUST
MATH 014 Calculus II
Final Examination Name:
30th May 2011 Student I.D.:
8:30-11:30 Tutorial Section:

1.
This is a closed book and notes examination.

2.
Do not open the exam until instructed to do so.

3.
When instructed to open the exam, please check that you have 12 pages of questions in addition to the cover page.


4.
Write your name, and other information in the space provided above.

5.
Show an appropriate amount of work for each problem. If you do not show enough work, you will get only partial credit.

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 10
Q. 2 10
Q. 3 10
Q. 4 10
Q. 5 10
Q. 6 8
Q. 7 10
Q. 8 12
Q. 9 9





Q. 10
Total Points
1. ([10 pts]) Consider the region bounded between the graphs ofy = f(x)and y = g(x)as shown in

the given .gure.
y
10 y = f(x)

y = g(x)
10 20x
(a)
Use the midpoint rule with n = 5 subintervals to estimate the area of the given region. [3 pts]

(b)
If the given region is rotated about the y-axis to obtain a solid of revolution, express the volume of the solid as an integral. [3 pts]

(c)
Use thetrapezoidal rule with n =5subintervalstoestimatethe volume ofthe solid obtained inpart(b). [4pts]


2. ([10 pts]) For a particular constant k, the following non-negative function f can be used as a probability densityfunctionforcertain randomvariable(randomnumber) X, i.e., the areabetween
the graph of f and the x-axis is 1.
.
kx
f(x)= . . (1+ 3x2)2 if x 0, .
.
. 0 if x < 0,

(a) Find the constant k. [5 pts]
y
(b)
Using the constant k foundinpart(a), .ndtheprobabilitythattherandomvariable X is lying between 1 and 3, i.e., the probability P{1 X 3}. [2 pts]

(c)
If F is the function de.ned by F(t)= P{X t}, with k asfoundinpart(a), sketch roughly the graph of F. [3 pts]


3. ([10 pts])Consider the parametric curve given by x =2t3 , y =3t2, where 0 t 5.
(a) Find the arc length of the parametric curve. [6 pts]
y
100

x
100 200 300
.100
(b)
Express the area of the surface of revolution obtainedby rotating thegiven curve about the x-axis by an integral. Do not evaluate the integral. [2 pts]

(c)
Express the area of the surface of revolution obtainedby rotating thegiven curve about the vertical line x = .2 by an integral. Do not evaluate the integral. [2 pts]


4. ([10 pts])Apply integration by parts to answer the following questions.

2
(a) Evaluate the integral x 4 lnxdx. [5 pts]
1

(b) Showthatif f is a function satisfying f(1)=3, f(1)=2, and fiscontinuous ontheinterval
1
[0,1] with |f(x)| 4