Math 146
H. Jordon
Section 7.7: Approximate Integration
b
Sometimes, its impossible to nd the exact value of a denite integral a f (x) dx. It may be dicult, or
even impossible, to nd an antiderivative, or it may be that the function f is determined from c
Calculus II
MAT 146
Integration Applications: Average Value of a Function
Here we apply integration to the task of determining the average value of a
function for a specified interval of independent-variable values. These ideas
extend what we typically th
MAT 146
Applying Differential Equations (9.3)
Newtons Law of Cooling
Now that we have techniques for approximating or actually determining solutions to differential
equations, we consider using those methods to solve
MAT 146
Direction Fields: Graphical Representations for Differential Equations
9.2
If we cannot analytically determine a solution to a differential equation, what options do
we have for generating a solution? Similar to
MAT 146
Introduction to Differential Equations
9.1
What is a Differential Equation? Also known as DIFEQ or D.E.
An equation in which a rate of change is expressed in terms of an independent and
dependent variabl
Math 146
H. Jordon
Section 7.6: Integration Using the Table of Integrals
In the back of the book, you will nd a Table of Integrals. These reference pages contain 120 rules for
integration, and often, we need to make a substitution or perform an algebraic
Math 146
H. Jordon
Section 6.4: Work
Work means the total amount of eort required to perform a task. In physics, the work W to
move an object is equal to the amount of force F required ( a push or pull) times the distance d
the object moves, i.e.,
W = F d
Math 146
H. Jordon
Section 5.5: The Substitution Rule
The substitution rule is an integration technique based on the chain rule. Suppose we want to integrate
f (x)f (x)dx
where f (x) is some function in x. Then, making the substitution u = f (x), we have
Math 146
H. Jordon
Section 7.4: Partial Fractions
Given
p(x)
where p(x) and q (x) are polynomials:
q (x)
Step 1: Divide q (x) into p(x) if the degree of p(x) is greater than or equal to the degree of q (x).
Step 2: Factor q(x) into linear factors (ax + b)
Math 146 H. Jordon
Section 6.1: Area between Two Curves
b
You have already seen that if f is a positive function on [a, b], then a f (x)dx gives the area under
the curve y = f (x) bounded by the x-axis and the lines x = a and x = b. What if we have two
fu
Math 146
H. Jordon
Section 6.3: Volume by Cylindrical Shells
We have already learned several methods for determining the volume of a solid: by disc, by washer,
or by slicing a solid into layers. In this handout, you are going to learn another method, call
Math 146
H. Jordon
Section 7.1: Integration by Parts
Integration by parts is an integration technique based on the product rule. Suppose we have functions u
and v of x. Then, we know
d
dv
du
uv = u
+v
dx
dx
dx
or
(uv ) = uv + vu
or
uv = (uv ) vu .
Thus,
u
Calculus II
MAT 146
Additional Methods of Integration: Using Trig Identities
The methods of substitution and integration by parts are widely used
methods of integration. Each of these methods is associated with a derivative
rule. Substitution relies on un