Section 6.2
Volumes
Volume for a cube:
Volume l w h
Area for a square
Area l w
Area for a circle
Area
r2
Volume for a cylinder
Volume
r2 h
Volume:The Disk Method
Finding area involving curves is one application of
integrals. Another application of integra
Section 8.3
The Integral Test and Comparison Tests; Estimating Sums
Testing convergence.
Theorem: The Integral Test
If f is positive, continuous, and decreasing for x
a n and
n 1
1
1 and a n
f n , then
f x dx
either both converge or both diverge.
Theorem:
Section 5.7 Part I
Additional Techniques of Integration
Trigonometric Integrals
Guidelines for Evaluating Integrals Involving Powers of Sine and Cosine
1. If the power of the sine is odd and positive, save one sine factor and convert the
remaining factors
Section 8.5
Power Series
Some functions can be represented exactly by an infinite series called a power series.
Definition of Power Series
If x is a variable, then an infinite series of the form
anxn
a1x1
a0
a2x2 . . . . . anxn . . .
n 0
is called a power
Math 182 CRN 22237 Course Outline
Fall 2013
We will complete the study of the major ideas of single-variable calculus begun in MA 181, selecting an
assortment of applications along with theory. The exact sections we cover may vary somewhat from what is
li
Montgomery College-Rockville
Department of Mathematics
Fall 2013
Math 182
Calculus 2
CRN 22237
6:30-8:45 pm
TR
127 Science West
Instructor: Prof. Chris Henrich
Office Hours: I will be available for additional questions for about a half-hour before class i
Math 182
CRN 22237
Homework 1
Due September 26, 2013
5 points
1
Name_
Show all work!
1. Approximate cos x 2 dx using the Midpoint Rule and n = 4. Write out the terms of the sum, then finish
0
the calculation.
2. Estimate the error in your approximation
AMATYC Student Math League
Math Competition
th
November 7
Training sessions every Thursday 2-3pm in
Science West 122, starting September 26th
Its free to train and compete, and no registration is
required
For more information
AMATYC Mathematics Competition info
The competition is open to students at two-year colleges nationwide, and is
broken down by region (we are in the Mid-Atlantic region).
Questions are at the level of Pre-Calculus Mathematics or lower, so all
Montgomery
Review from Calc 1
You should know the basics from Calc 1 very well: derivatives of basic functions, product rule,
quotient rule, chain rule. From reference page 5 in the back of the book, know #1-9, 11, 13-15
without looking; be familiar with 10, 12, 16-
Section 5.5
So far, we have only done integrals of simple functions.
For integration, there is NO product rule, NO quotient rule, NO chain rule.
Examples:
t1 t dt
2
Multiply out the polynomial, then integrate.
x -1
dx
x
Use laws of exponents to rewrite w
Section 5.7
Trig integrals
odd powers of sine and cosine:
Examples:
sin
3
x dx
We did integrals like this in the previous section using a reduction formula, but lets try a more
generally applicable method. We cant use u = sin x, du = cos x dx, because we
Section 4.5
To prepare for section 5.10, we are going back to revisit some types of limits that we couldnt
evaluate before (in calc. 1.) First lets look at some familiar limit situations:
If a function is continuous at x = a, then lim f(x) f(a) . If a fun
Section 5.8
Integration using tables:
Find a formula on the table (reference pages 6-10 in the back of the book) that matches or is
similar in form to a given integral. Use substitution or rewrite algebraically to make an exact
match. Use the formula to g
Although the same study methods dont work for everyone, if you find that what you are doing
now is not as effective as you would like, here are some suggestions of things to try.
If you can, look over the class notes or the section in the book before we c
Section 5.10
b
The definition we have used of a definite integral f(x)dx required a finite interval [a, b] of
a
x-values and an integrand, f(x), that has no infinite discontinuities on [a, b].
Does it make sense to talk about integrals on infinite interva
Section 6.4 continued.
Parametric Equations
Here is some background on parametric equations for
those who dont have any experience with them.
Plane Curves and Parametric Equations
Instead of looking at a graph represented by a single
equation with two var
Chapter 8
Infinite Sequences and Series
Section 8.1
Sequences
Mathematically a sequence is defined as a function whose domain is the set of positive
integers. Instead of using standard f x notation for sequences, we use a n . The n
represents the number o
/// //l//
%4¢? yié . . 1
Calculus II 9" Test 1 Name WXDXQSAkSdq
l. Simpsons formula for the approximation Sn of a denite integral f f (x)dt 15 FM
as follows:
%[f(x°)+4f(xl)+2f(xz)+4f(x3)+~--+2f(x,,-2)+4f(xn_1)+f(xn)l,
ba
n .
' where Ax =
4
Class Notes
Section 5.4
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus, Part 1
If f is continuous on a, b , then the
function g defined by
x
gx
f t dt a x b
a
is an antiderivative of f, that is, g x
f x for a
x
b.
1) Find the deri
Chapter 7
Differential Equations
Section 7.1
Modeling with Differential Equations
In the real-world, mathematical models come from intuitive reasoning about something or
from laws based on experiments. Often these models take on the form of differential
e
Section 7.3
Separable Equations
Consider the differential equation that can be written in the form
M x N y dy
0
dx
where M and N are continuous functions of the variables x and y.
Example
Our goal is to get the everything related to each respective variab
Class Notes
Section 5.7 continued.
Partial Fractions
In this section, we will break down singular rational expressions into multiple rational
expressions. The goal is to have rational expressions here we can apply some of the basic
integration rules. This
Section 5.6
Integration by Parts
Theorem: Integration by Parts
If u and v are functions of x and
have continuous derivatives, then
udv uv
vdu.
Guidelines for Integration by Parts
1. Try letting dv be the most
complicated portion of the integrand
that fits
Section 6.3
Volumes by Cylindrical Shells
The shell method is an alternative to using the disk method to find the volume of a solid.
The Shell Method
The formulas for finding the volume using the shell method are as follows:
Horizontal Axis of Revolution