Math 132 Section 6.1 Areas Between Curves
Finding the Area Between Curves:
Area =
Examples:
Ex 1: Find the area between the two curves in the graph below:
1
2
Ex 2: Find the area between = 2 6 and = 12 2 .
3
Ex 3: Find the area between = cos() and = 1 cos
Math 132 Section 7.4 Integration by Partial Fractions
1
To integrate rational functions, we can break the function into a sum of simpler fractions that we
can integrate.
()
Rational Function: ( ) = ()
To use the method of partial fractions, the rational f
Math 132 Section 7.1 Integration by Parts
1
Integration by Parts
Product Rule
When integrating two functions multiplied together, and there is no relationship
between the two functions, we use the method of integration by parts.
Integration by Parts:
()
Math 132 Section 5.3 The Fundamental Theorem of Calculus
1
The Fundamental Theorem of Calculus has two parts. Suppose f is a continuous
function on the closed interval [a,b].
I.)
If () = () then () = ()
II.)
() = () (), where F(x) = f(x)
~
Part II: () =
Math 132 Section 6.2 Volumes
1
Disk Method:
Volume = ( ) or ()
2
Examples:
Ex 1: Find the volume of the solid obtained by rotating the region
= from = 0 to = 1 about the x-axis.
3
Ex 2: Find the volume of the solid obtained by rotating the region
= 1 2
Math 132 Section 7.2 Trigonometric Integration
1
When integrating trigonometric functions involving sine and cosine, look for the following:
Pattern
Step 1
Identity
u-substitution
Odd Cosine
Factor out one power
of cosine
cos 2 () = 1 sin2()
= sin()
Odd
Olivia Flynn
Calculus II
Sections 5.3-5.5, 6.1-6.2, 7.1-7.3
October 10 & 11 reviews 7-9pm
Midterm #1 Wednesday, October 12th, 7-9pm in Goessman Room 64
Old MT #1 posted on the course chairs blog (no solutions)
Derivatives
d
sin (x)=cos( x)
dx
d
cos(x )=si
Olivia Flynn
Calculus II
Section 7.4 - Integration of Rational
Functions By Partial Fractions
Note: for any k, you should be able
to write down the partial fraction
decomposition. For small k you
should be able to solve for
coefficients.
Long Division of
Practice Integrals
Math 132
R
1. (ln x)2 dx
R
2.
x sin3 x dx
R
3.
x3 1 x2 dx
R ln y
4.
5.
6.
7.
8.
9.
10.
dy
y
R
R
R
R
R
R
sin5 x cos4 x dx
(ln x)2
dx
x3
2
dx
2
2x + 3x + 1
cot2 x dx
x2
dx
1 x2
sin x + sec x
dx
tan x
R
11.
tan5 x dx
R sin x cot x
12.
13
Trigonometric Identities and Integrals
Erica M. Farelli
Math 132 - University of Massachusetts
Trigonometric Identities
1
1
sec x =
csc x =
sin x
cos x
tan x =
sin x
cos x
sin2 x + cos2 x = 1
cot x =
R
R
R
R
R
R
R
R
1 + cot2 x = csc2 x
cos 2x = cos2 x sin
Mutiple Choice Section: Choose the one option that best
answers the question. There is "no partial credit for questions
1-5.
1. [5 points] Which of the following integrals calculates the
area of the shaded region?
re) y (x3
r at ' - 1-.-
III]. II.
I'd
The velocity function (in meters per second) for a particle moving along a line is given by
v(t)=3t5,0t3.
(a) Find the displacement (in meters) of the particle.
Displacement = meters
(b) Find the total distance traveled (in meters) by the particle.
Total
MATH132 Spring 2017 Worksheet: Fundamental Theorem of Calculus
1) Graph the two curves
y = 3/2x 1/2
y = 3/4x2 11/4.
Set up the definite integral to find the area enclosed by these two curves
2) Find the derivative of the following functions
Z x
1 + 7t2 si
Math 132 Section 11.3 Integral Tests
Integral Test:
Suppose is a continuous, positive, and decreasing function on [1, ) and let = ().
Then,
P-Series:
=1
1
Examples: Do the following series converge or diverge?
1
1
1
1
Ex 1: 1 + 8 + 27 + 64 + 125 +
1
2
1
Trigonometric Identities and Integrals for Chapter 7
Math 132
Trigonometric Identities
csc x =
1
sin x
sec x =
tan x =
sin x
cos x
cot x =
sin2 x + cos2 x = 1
1
cos x
cot x =
cos x
sin x
tan2 x + 1 = sec2 x
sin x cos x = 21 sin 2x
sin2 x = 12 (1 cos 2x)
1
Lecture 17: Power series.
A power series about a, or just a power series is
X
cn (x a)n ,
n=0
a is called the center, cn s are called the coefficients of the power series. By convention:
(x a)0 = 1,
even when x = a.
Two new concepts:
1. Radius of converge
Lecture 10: Improper Integrals
1
Improper Integral of Type I: When interval is infinite.
1.
R
f (x)dx = limt
Rt
f (x)dx.
Rb
Rb
2. f (x)dx = limt t f (x)dx.
R
Rc
R
3. f (x)dx = f (x)dx + c f (x)dx, for any number c, c can be chosen to be 0
for simplicity.
Lecture 13: Integral test and Comparison test
Integral Test.
THE INTEGRAL TEST: Suppose
P f is a continuous, positive, decreasing on [1, ) and
let an = f (n). Then the series n=1 an is convergent if and only if the improper integral is
convergent.
R In ot
Lecture 11: Sequences
Definition and notation for sequences.
A sequence is a list of numbers written in certain order: a1 , a2 , a3 , , an , , where a1 is
called the first term, an is called the n-th term or general term of the sequence.
There are two way
Lecture 12: Series
Definition and notation for series.
Given a sequence cfw_an , the associated series is created by taking the summation of all
elements in the sequence.
P
Formally, the sequence is denoted by
n=1 an .
To give a precise definition, first
Lecture 18: Representing functions as power series.
At this point, we only know the following power series representation:
1
= 1 + x + x2 + x3 +
1x
X
=
xn , for |x| < 1.
n=0
We now look at three techniques allow us to obtain a series representation using
Lecture 9: Integration by partial fractions
In this section we are going to take a look at integrals of rational functions
Z
R(x)dx
(1)
where
R(x) =
P (x)
.
Q(x)
(2)
both P (x) and Q(x) are polynomials.
The process of taking a rational expression and deco
Lecture 8: Trigonometric Substitution
1
Case 1:
a2 x2
Make a substitution x = a sin(), then dx = a cos()d and
C
a
A
a2 x2 = a cos().
x
a2 x2
B
1. Evaluate the integral
Let x = 5 sin(), then
R
25x2
x2
Z
.
dx.
25 x2
dx =
x2
Z
5 cos()
5 sin()d
25 sin2 ()
Z
Math 132 Section 11.1 Sequences
1
Sequence: a list of numbers written in a definite order.
Notations:
1 , 2 , 3 , , ,
cfw_
cfw_
=1
Examples: Write out the first five terms of the following sequences.
Ex 1: cfw_+1
=1
Ex 2: cfw_
(1) (+1)
3
=1
Given the
Math 132 Section 11.2 Series
Any decimal can be written as a series.
Write = 3.14159265 as a series.
=
Some series converge and others diverge.
Convergent Series:
=1
1
=
7
Divergent Series:
=2
2
=
1
1
2
Geometric Series:
1 = + + 2 + 3 +
=1
Ex 1: Does th
Sequences and Series Convergence Tests
Math 132
Sequences
A sequence is increasing if an < an+1 for all n 1.
A seqeunce is decreasing if an+1 < an for all n 1.
A sequence is called monotonic if it is either increasing of decreasing.
A sequence is boun
Math 132 Section 11.4 The Comparison Tests
Comparison Test:
Suppose and are series with positive terms.
1.) If is convergent and for all , then is also convergent.
2.) If is divergent and for all , then is also divergent.
Ex 1: Does the series converge or