Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 5: Lengths of Curves
Definition (The First Arc Length Formula for = () ):
Definition (The Second Arc Length Formula for = () ):
Let have a continuous first order derivative on interval
Consider the following
f?77
where
(Ja x Y
.
Consider the following
integral:
Z
l
.77dX
Z
a> O.
where
? 7
J
7 . ( a Z
integral:
+ x Z )'
Consider the following
l
7 7 dx
a> O.
? 7
J
7 . ( x Z

integral:
y .
a2
7 7 dx
a> O.
where
Motivation 1:
How do we ev
Definition (Area of Regions in Polar Coordinate System 1):
Let region
R
e=a
e = f3 . Then the area of the region
and
r
be bounded by the graph of
Area =
tce) ~
0 between two rays
R is given by
1
8=fJ
r
=
C tee)2 de
J8=a
2
An Explanation:
We are going to
Definition of Antiderivatives:
If we are given two functions [(x)
that F(x)
and F(x) satisfying the following condition: F' (x)
is an antiderivative of [(x)
cfw_(x)
F(x)
I
[Differentiation
= F;(x) = F;(x) =
F~(x)
Besides, all antiderivatives
an appropriat
()= rr/2
;:;:i~;~<;,~i\
In rectangular coordinate system, we use a rectangular net to
describe and capture points, curves on the plane. In this
reference system, points are sometimes called rectangular
points and curves are called rectangular curves.
In p
If we are given a rectangular equation which can define a realvalued differentiable function of a single variable, for instance
y = f(x) = Xl , then there must be a unique tangent line to the given curve at a given point. For example, the slope of the
ta
~\'
f'cfw_(;n"it
_lV+',
~I(
,I
3X2 + 7x  2
dx
x3  x2  2x
f
An initial
preparation: Stage 1: The giVen integral does not belong to anyone of the basic indefinite integrals introduced in
Lecture Notes 9, page 1. Stage 2: Addition, subtraction rule for
Techniques/methods involved: various differentiation rules: +, rules,
product, quotient rules, power rule, chain rule, ., etc
.~(
) Use .'2
;.:_a_x to find slope of tangent lines to the curve of y = F(x) and SDon . ~
.
.
I
Definition
ir7 [(x,
_t
y
.
~th
1014Calculus II (Written by Dr. HonMing HO)
/
~
Lecture Notes 19: Improper Integrals II
Example 1: (This example shows that there is a solid having finite volume but infinite surface area. This solid is called Gabriel's
Horn.) Let R be the region bo
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Math1014 Midterm Exam, Spring 2014
Part I: Multiple Choice Questi
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HKUST
MATH1014 Calculus II
Midterm Examination (White Version)
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HKUST
MATH1014 Calculus II
Final Examination (White Version)
Name:
Math1014 Midterm Exam Solution, Spring 2013
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Part I: MC Answers
Whi
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1
Math1014 Final Exam, Spring 2013
MC Answers
White Version
Questi
R about the xaxis, where the region R is
Example 1: Find the volume of the solid that is generated by rotating a region
bounded by the graph of y = sin2 x over the interval [0, rr] .
An initial preparation:
When you try to solve this question, you think
L.I\ '
J
dv = uv 
U
J
f
v du
Integration
ltv
]~ 
x=a
J
Initial preparation:
x=b
u dv = [
f
x=b
V
du
x=a
x eX dx.
is the reverse process of differentiation.
equivalent to finding (or sometimes guessing) F(x)
J
S', ~r'1'1"l
d
such that

dx
[F(x)
]
Inte
Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 7: Work Done by Force
Exercise 1:
A bag of sand originally weighing 144 lb was lifted at a constant rate. As it rose, sand also leaked out at a constant rate (lb per ft).
The sand was h
Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 3: Volumes by Slicing, Volumes of NonRevolving Solids
Exercise 1:
The base of the solid is the disk 2 + 2 1 . The crosssections cutting by planes perpendicular to the axis between =
Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 9: Trigonometric Integrals
Exercise 1:
Find the length of the curve = lncos  where 0
3
.
Solution to Exercise 1:
1) Things to do: There are two arc length integrals we may apply in t
Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 4: Volumes of Solids of Revolution
Exercise 1:
Find the volumes of the solids generated by revolving the regions bounded by the following lines and curves about the axis:
a)
= cos , 0
Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 2: Areas of Regions between Two Curves
Exercise 1:
Find the area of the yellow region which is bounded above by the blue curve = 2 2
and bounded below by the red curve = 4 2 2 on the ri
Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 6: Areas of Surfaces of Revolution
Definition (the 1st formula for area of a surface of revolution
about the axis):
Definition (2nd Formula of Area of a Surface of Revolution
about the
Math 1014Calculus II (Written by Dr. HonMing HO)
Practice Exercises 8: Integration by Parts
Exercise 1:
Evaluate the following indefinite integral:
2 .
Initial preparation: Integration is the reverse process of differentiation. Integrating 2 with respe
Graph of y = [ex)
rex)
Definition (The First Arc Length Formula for y =
I:
Let [ have a continuous first order derivative on interval
[a, b] . The arc length of the curve from point (a,[(a)
to
(b,[(b) is given by
~
X=b
j
L=
x:a
[a, bJ ?
[a, b] into n
What
i
YV
.
y
Definition of area between two curves with respect to x:
Suppose that two continuous functions satisfy g(x) 2': rex) for
every x in interval a:; x :;b . The area of the region bounded
by the curves of 9 and [ on the interval a:; x :;b is defined
Example 1: Find the arc length of the astroid with equation X2/3
+ y2/3
= 1.
Solution to Example 1:
1)
Things to do:
Decide which arc length formula we may apply. We observe
the graph of the given equation exhibits symmetry. From the equation, it is
not d
Suppose
that

d
dx
[F(x)]
:=:
[(x)
.
Function F(x) is called an antiderivative of
cfw_(x)
There
are
infiniteiy
mSlny
antiderivatives
of cfw_(x).
The geometrical ~
meaning of F(x)
is that it is the area
producing funclion of [(x) .
y
Given
.
Differentiati
Definition (Volume of a Solid ObjectGeneral Slicing Method II ):
Suppose that a solid object extends from y = c to Y = d and the
crosssection of the solid perpendicular to the yaxis has an area
given by a function B that is integrable on [c, d] . The v