M408S
Integral Calculus and Series
Fall 2010
MWF 8:00 to 9:00 in ECJ 1.202
(55115, 55120, 55125)
INSTRUCTOR: Dr. Gary Berg
Office: RLM 13.164 phone: 471-6410
email: gberg@math.utexas.edu
Office Hours: MWF 1 to 2:30 pm - or by appointment.
TA: James Delfel
Calculus II-Stewart
5.1
Dr. Berg
Spring 2010
Areas and Distances
Integral calculus arose from the need to calculate areas and volumes of complex
shapes.
The Area Problem
Before the Fundamental Theorem of Calculus was discovered, the standard
method of cal
Calculus II-Stewart
5.4
Dr. Berg
Spring 2010
Indefinite Integrals and the Net Change Theorem
Antiderivatives play such an important role in integration that integral notation is
commonly used to represent them.
Definition
f ( x ) dx = F ( x ) means
d
dx
F
Calculus II-Stewart
6.2
Dr. Berg
Spring 2010
Volume
Volume by Cross Section
A right cylinder has congruent cross sections and side(s) at right angles to the
base. A prism, for example, is a right triangular cylinder. If the area of a cross section is A
an
Calculus II-Stewart
6.1
Dr. Berg
Spring 2010
Area Between Curves
To approximate the area between two curves, we can use rectangles as we did
before. Suppose that f ( x ) g( x ) on the interval [ a, b] . Then, if we divide the interval
ba
into n intervals
Calculus II-Stewart
5.5
Dr. Berg
Spring 2010
The Substitution Rule
The substitution rule reverses the chain rule. Recall that, by the chain rule,
[ f (g( x )] = f ( g( x ) g( x ) . Reversing this yields
d
f (g( x )g( x ) dx = dx [ f (g( x )] dx = f (g( x
Calculus II-Stewart
7.4
Dr. Berg
Spring 2010
Logarithmic Functions
Derivatives
Theorem
d
dx
1
1
d
(ln x ) = x and dx (ln g( x ) ) = g( x ) g( x ) .
Example A
d
dx
1
(ln (tan x ) = tan x sec 2 x .
Example B
d
dx
(ln x
3
)
2x =
1
(3x 2 2) .
x 2x
3
Antideri
Calculus II-Stewart
7.2
Dr. Berg
Spring 2010
Calculus with Exponential Functions
The Exponential Function
Bacteria double their population at regular intervals through cell division. We
speak of the half-life of radioactive material. Exponential functions
Calculus II-Stewart
7.6
Dr. Berg
Spring 2010
Inverse Trig Functions
Inverses
The inverses of the trig functions are an important addition to our toolbox. Recall
that f 1 ( x ) = y if and only if f ( y ) = x , and that to be invertible, a function must be