5.1: Areas and Distances
5.2: The Denite Integral
(Dated: September 21, 2011)
THE AREA PROBLEM
Part of the area problem is to make the intuitive idea of
what the area of a region is precise by giving it an exact definition of area. This is easy for a regi
9.1: Modeling with Differential Equations
(Dated: November 8, 2011)
Roughly speaking, a differential equation is an equation
that contains an unknown function and some of its derivatives. The order of the equation is the order of the highest
derivative th
Math 152 Au. 11 K. Kwa
Name:
SO N0056
Signature:
October 27, 2011
Name.nnn:
Exam 2
Form A, 7 pages
General Instructions:
Do all the problems. Answer each part thoroughly.
Show all of your work! Incorrect answers with work shown may receive partial credit
List of Paper Homework to be Turned In for Math
152.01 (Autumn 2011)
Kiam Heong Kwa
November 8, 2011
Sec 9.1 due 12/01: 2, 4, 8, 12
Sec 8.2 due 11/15: 6, 12, 14, 26, 28, 30
Exercise 30: The upper half of the torus is generated by rotating the curve (x R)2
Math 152 Sp. 10 Z. Fiedorowicz
Name:
EA 170
Signature:
April 14, 2010
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Exam 1
TA:
Suh Yeong Park
Form A, 7 pages
Rec. Time:
12:30
Xiaoyue Xia
1:30
General Instructions:
Do all the problems. Answer each part thoroughly.
Show all of your work! Inc
October 25, 2011
Name
Name.nnn
TA & Rec. hour
Math. 152 Quiz 3 Form A
PROBLEM 1. (10 points) A force of 8 lb is required to hold a spring stretched 7 in. beyond its
natural length. How much work is done in stretching it from its natural length to 9 in bey
8.2: Area of a Surface of Revolution
(Dated: November 8, 2011)
THE SURFACE AREA OF A FRUSTUM OF A CIRCULAR CONE
Consider a circular cone with base radius r and slant
height l . It can be attened to form a sector of a circle with
2 r
2 r
radius l and centr
8.1: Arc Length
(Dated: November 8, 2011)
Writing f in the Leibni notation, one has
THE ARC LENGTH FORMULA
Let C be a curve whose dening equation is y = f (x ),
where f is a continuous function on [a , b ]. A polygonal
approximation to C can be obtained b
5.3: The Fundamental Theorem of Calculus
(Dated: September 23, 2011)
Let f be integrable on [a , b ]. Then it makes sense to dene the function
Notation:
F (b ) F (a ) = F (x )]b = F (x )|b = [F (x )]b .
a
a
a
x
g (x ) =
f ( t ) d t , a x b.
a
If f is posi
5.4: Indenite Integrals and the Net Change Theorem
(Dated: September 23, 2011)
Let f be a function on the interval [a , b ]. Recall that an antiderivative of f is a function F such that F = f . The traditional notaion to denote this statement is F (x ) =
5.5: The Substitution Rule
(Dated: September 25, 2011)
ex
d x Hint:
ex + 1
cos(/x )
d x Hint:
x2
35*:
f (g (x )g (x ) d x =
32*:
34*:
Theorem 1 (The Substitution Rule for Indenite Integrals)
If u = g (x ) is a differentiable function whose range is an
int
6.1: Areas Between Curves
(Dated: September 26, 2011)
The area A of the region bounded by the curves y = f (x ),
y = g (x ), and the lines x = a and x = b , where f and g are
continuous is given by
16*: y = x 3 x, y = 3x Hint:
when x
=
0, 2.
The curves in
6.2: Volumes
(Dated: October 1, 2011)
A (right) cylinder is a solid bounded by two plane congruent regions, say B 1 and B 2 , lying in parallel planes. It is
customary to call one of these planes, say B 1 , the base of
the cylinder. The height of the cyli
6.3: Volumes by Cylindrical Shells
(Dated: October 5, 2011)
Consider a cylindrical shell with inner radius r 1 , outer radius r 2 , and height h . Its volume is
2
2
V = (r 2 r 1 )h
6.3.4*: y = x 2 , y = 0, x = 1 Ans:
2
1
0
2 x 2
1
dx =
0 2 x e
2 x x 2 d x
6.4: Work
(Dated: October 7, 2011)
Let x (t ) be the position function of an object of mass m
moving along a straight line as a function of time t . Then
the force f on the object can be calculated from Newtons
second law:
f (t ) = m
d 2x
.
dt2
In the SI
6.5: Average Value of a Function
(Dated: October 10, 2011)
Let f be a function on an interval [a , b ]. Dividing [a , b ]
ba
and choosing
into n subintervals of equal width x =
n
a sample point from each interval yields the average of the
numbers f (x 1 )
7.1: Integration by Parts
(Dated: October 12, 2011)
Let f (x ) and g (x ) be differentiable functions. Then the
product rule states that
Then, by integration by parts also,
sin x
x sin x
dx
x sin x cos x
=
+ C1.
+
2
x cos x d x =
d
f (x )g (x ) = f (x )g
7.2: Trigonometric Integrals
(Dated: October 15, 2011)
STRATEGY FOR EVALUATING sinm x cosn x d x ,
WHERE m 0, n 0 ARE INTEGERS
Example 2 (Exercise 7.2.3 in the text)
sin5 x cos3 x d x =
1. If the power of cosine is odd (n = 2k + 1), save one cosine factor
7.3: Trigonometric Substitution
(Dated: October 20, 2011)
To make the trigonometric substitutions x = a sin , x =
a tan , and x = a sec , one needs to restrict such that x =
x ( ) is one-to-one. Also, these trigonometric substitutions
are effective for di
7.4: Integration of Rational Functions by Partial Fractions
(Dated: October 22, 2011)
Recall that a (real) polynomial is a function of the form
A rational function is a function of the form
f (x ) =
n
P (x ) =
a i x i = a n x n + a n 1 x n 1 + + a 1 x + a
7.8: Improper Integrals
(Dated: October 29, 2011)
In conclusion,
IMPROPER INTEGRALS OF TYPE 1
(a) If
t
a
f (x ) d x exists for all t a , then
t
1
f (x ) d x := lim
f (x ) d x
t a
a
provided the limit on the right exists as a nite number. Here and hereafte
November 8, 2011
Name
Name.nnn
TA & Rec. hour
Math. 152 Quiz 4 Form A
PROBLEM 1. (10 points) Write down the general form of the partial fractions decomposition of
(x 17)9
(x4 + 16x2 )(x2 11x + 30)2
Do NOT determine the numerical values of the coecients.
S