Math 1452 Exam 1 Review
Test 1 is on Monday, Sept 26 and it covers sections 6.1 6.5(moments and center of mass only)
Sec 6.1 Area Between Curves
1. Find the area of the region in the first quadrant, between the y-axis and the graphs of
1
y= xx + y=2
Ans:
CHAPTER 8: INFINITE SERIES
8.1: Sequences and Their Limits
Definition. A sequence cfw_an is a function whose domain is the set of nonnegative integers and whose range is a subset of the real numbers. The
functional values a1 , a2 , a3 , are called the te
CHAPTER 8: INFINITE SERIES
Section 8.5: The Ratio Test and the Root Test
Theorem (Theorem 8.16 (The Ratio Test). Given the series
ak > 0 for all k, suppose that
P
ak with
ak+1
= L.
k ak
lim
The ratio test states
Pthe following:
(1) If L < 1, then P ak con
CHAPTER 7: METHODS OF INTEGRATION
7.7: Improper Integrals
In this section we discuss the two types of improper integrals. Improper integrals
of the first type are integrals where one, if not both, of the limits of integration
are infinite values. We are m
CHAPTER 7: METHODS OF INTEGRATION
7.3: Trigonometric Methods (Part 1)
This section is one of the biggest in Calculus 2, so we must take care to work
as many problems as we can. There are two main parts to this section. The
first part gives us a process to
CHAPTER 8: INFINITE SERIES
Section 8.2: Introduction to Infinite Series; Geometric Series
Definition. An infinite series is an expression of the form
a1 + a2 + a3 + =
X
ak
k=1
and the nth partial sum of the series is
sn = a1 + a2 + a3 + + an =
n
X
ak .
k=
CHAPTER 7: METHODS OF INTEGRATION
7.2: Integration by Parts
Definition (Integration by Parts). Let u and v be functions of x. Then,
Z
Z
u dv = uv v du.
The above formula is the formula for integration by parts. If we have an
integral in which we want to u
CHAPTER 6: ADDITIONAL APPLICATIONS OF THE INTEGRAL
6.2: Volume
Volume of a Solid with Known Cross-Sectional Area: A solid gure S with a
crosssectional area A(:c) perpendicular to the Xeaxis at each point on the inter
val [(1, b] has volume
in
Volume 2 f A
CHAPTER 7: METHODS OF INTEGRATION
7.4: Partial Fraction Decomposition
Definition (Partial Fraction Decomposition (Linear Factors). Let f (x) =
P (x)
(xr)n , such that P (r) 6= 0 and the degree of P (x) is strictly less than that
of n. The f (x) can be dec
CHAPTER 9: VECTORS IN THE PLANE AND IN SPACE
Section 9.1: Vectors in R2
Denition. A vector is a quantity (such as velocity or force) that has both
magnitude and direction. We denote the vector v by a bold lower case letter,
in when we are typing. When w
CHAPTER 7: METHODS OF INTEGRATION
7.3: rIrigonometric Methods (Part 2)
In this part of section 7.3, we concern ourselves with integrating functions in
volving V0.2 m2, V322 a2, and mi? + 9:2. It is important for us to check that
integration by substitutio
CHAPTER 6: ADDITIONAL APPLICATIONS OF THE INTEGRAL
In this nal part of section 6.5, we focus on liquid force.
Denition. Consider a at surface (or a plate) submerged vertically under a
llqald with a weight density given by p. If the plate is submerged from
CHAPTER 6: ADDITIONAL APPLICATION OF THE INTEGRAL
Section 6.5: Physical Applications: Work, Liquid
Force, and Centroids
In this part of section 6.5, we discuss centroids.
Denition. In mechanics, its often important to determine the point where an
irregula
CHAPTER 6: ADDITIONAL APPLICATIONS OF THE INTEGRAL
In this part of section 6.5, we focus on work.
Denition. If a body moves a distance d in the direction of an applied constant
force F, the work, W, done is
W = Fcl.
Example 1. Find the amount of work done
CHAPTER 6: ADDITIONAL APPLICATIONS OF THE INTEGRAL
6.3: Polar Forms and Area
The main focus of this section is working in polar coordinates. In polar coordi
nates, the xaxis is called the polar axis and the yaxis is called the radial axis.
The origin, g
CHAPTER 8: INFINITE SERIES
Section 8.4: Comparison Tests
Theorem (Theorem 8.13 (The Direct Comparison
Test).
P
P
(1) Suppose 0 ak ck for all k and that
ck converges. Then
ak also
converges.
P
P
(2) Suppose 0 dk ak for all k and that
dk diverges. Then
ak a
CHAPTER 8: INFINITE SERIES
8.3: Integral Test; P-Series
Theorem (Theorem 8.9 (The Divergence Test). Given the series
limk ak 6= 0, then the series must diverge.
P
ak , if
This is a very important theorem, but it is important to know what it
says, and what
Derivatives
You are expected to have the following rules memorized. Assume that u is a function of x.
1. Product Rule: F S 0 + SF 0
Quotient Rule:
BT 0 T B 0
B2
2.
d n
u = nun1 u0
dx
3.
d
1
u = u0
dx
2 u
4.
u0
d
ln u =
dx
u
5.
d
u0
logb u =
dx
u ln b
6.
d
Sec 6.5
Class Examples
Center of Mass Examples
A homogeneous lamina has constant density = 1 and is bounded by the graphs given
M x =
M y =
below. Find the
moment about the x-axis, the
moment about the y-axis,
and the coordinates of the center of mass, (
6.3 Polar Forms and Area
Table 6.2 Directory of Polar-Form Curves
LtMAgONS - keb'iacueland r=bizasin9 '-
b , 7
r=bacos.a<I rabaoose,1<2
a
standard form, inner loop
standard form, dimple
r=aacost9 r=a+acos
standard form If rotation
ROSE CURVES
r = acos
Webwork A3
#11
b
Polar form of arc length integralis s= r 2 +
a
Here,
r=sin+ cos
dr 2
d
d
( )
dr
=cos sin
d
s= ( sin +cos )2+ ( cos sin )2 d
0
=
( s 2 +2 sin cos +cos2 )
+( s 2 2 sin cos +cos2 ) d
0
2
2
s +cos
2()
=
=
2 d= 2
0
( 2 s 2 +2 cos 2 ) d=
Chapter 6 Formulas
Area between two curves =
or
d
b
[top curve a
bottom curve]dx if you are using vertical strips
[right function - left function]dy if you are using horizontal strips
c
Volume of a solid with a
CHAPTER 6: ADDITIONAL APPLICATIONS OF THE INTEGRAL
6.4: Arc Length and Surface Area
Despite the name to this section, in this class, we are only going to cover arc
length.
Denition. A function ax) is said to be continuously (ii'erentieble on the
closed in
CHAPTER 6: ADDITIONAL APPLICATIONS OF THE DERIVATIVE
1 Area Between TWO Curves
Denition. Suppose f(:L') and 9(3) are functions on the closed interval [(1,1)]
such that f(:L') 2 9(5) for all .7; 6 [cc b]. Then. the area, between the two curves
is given by
CHAPTER 9: VECTORS IN THE PLANE AND IN SPACE
Section 9.3: The Dot Product
Definition. Suppose u = a1 i + a2 j + a3 k and v = b1 i + b2 j + b3 k. The
dot product of u and v, written as u v is given by
u v = (a1 b1 ) + (a2 b2 ) + (a3 b3 ).
Note that the dot
CHAPTER 9: VECTORS IN THE PLANE AND IN SPACE
9.4: The Cross Product
Definition. Suppose u = a1 i + a2 j + a3 k and v = b1 i + b2 j + b3 k. The
cross product, written u v is the vector
u v = (a2 b3 a3 b2 )i + (a3 b1 a1 b3 )j + (a1 b2 a2 b1 )k.
Some of you
CHAPTER 8: INFINITE SERIES
Section 8.6: Alternating Series; Absolute and Conditional Convergence
Theorem (Theorem 8.18 (The Alternating Series Test). The alternating series
X
(1)k ak
X
or
k=1
(1)k+1 ak
k=1
converges if the following two conditions hold:
(
CHAPTER 9: VECTORS IN THE PLANE AND IN SPACE
Section 9.2: Coordinates and Vectors in R3
For Calculus 2, we only focus on the vector portion of this section. Whats
more, we have all of the same properties for vectors in 3 dimensions as we do
in 2 dimension
CHAPTER 8: INFINITE SERIES
Section 8.7: Power Series
Definition. An infinite series of the form
X
ak (x c)k = a0 + a1 (x c) + a2 (x c)2 +
k=0
is called a power series in (x c). The numbers a0 , a1 , a2 , are called
the coefficients of the series.
Theorem
CHAPTER 8: INFINITE SERIES
Section 8.8: Taylor and Maclaurin Series
The ultimate goal of this section is to find polynomial approximations of
non-polynomial functions.
Definition. The nth -degree polynomial function of f at x = 0 is called
the nth -degree