Volume for a cube:
Volume l w h
Area for a square
Area l w
Area for a circle
Volume for a cylinder
Volume:The Disk Method
Finding area involving curves is one application of
integrals. Another application of integra
The Integral Test and Comparison Tests; Estimating Sums
Theorem: The Integral Test
If f is positive, continuous, and decreasing for x
a n and
1 and a n
f n , then
f x dx
either both converge or both diverge.
Section 5.7 Part I
Additional Techniques of Integration
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
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
a2x2 . . . . . anxn . . .
is called a power
Math 182 CRN 22237 Course Outline
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
Department of Mathematics
127 Science West
Instructor: Prof. Chris Henrich
Office Hours: I will be available for additional questions for about a half-hour before class i
Due September 26, 2013
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
2. Estimate the error in your approximation
AMATYC Student Math League
Training sessions every Thursday 2-3pm in
Science West 122, starting September 26th
Its free to train and compete, and no registration is
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
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-
So far, we have only done integrals of simple functions.
For integration, there is NO product rule, NO quotient rule, NO chain rule.
t1 t dt
Multiply out the polynomial, then integrate.
Use laws of exponents to rewrite w
odd powers of sine and cosine:
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
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
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
The definition we have used of a definite integral f(x)dx required a finite interval [a, b] of
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.
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
Infinite Sequences and Series
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
%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
' where Ax =
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
f t dt a x b
is an antiderivative of f, that is, g x
f x for a
1) Find the deri
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
Consider the differential equation that can be written in the form
M x N y dy
where M and N are continuous functions of the variables x and y.
Our goal is to get the everything related to each respective variab
Section 5.7 continued.
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
Integration by Parts
Theorem: Integration by Parts
If u and v are functions of x and
have continuous derivatives, then
Guidelines for Integration by Parts
1. Try letting dv be the most
complicated portion of the integrand
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