Notes from Calculus I. I created these notes to help get me through Calculus I. My notes end somewhere around the end of Volume 1 of the OpenStax Textbook, because that's all we used in class.
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Rules for Indefinite Integrals
Properties of Indefinite Integrals
Let and be antiderivatives of and , respectively, and let be
any real number.
Sums and Differences
Constant Multiples
Definition of Partitions
A set of points for with , which divides the interval into subintervals of the form , is called a partition of . If
the subintervals all have the same width, the set of points forms a regular
partition (or uniform partition ) of the interval .
So, to create regular partitions on a closed interval, we start by selecting
evenly spaced points along
And then index each boundary point between partitions.
Rule: Left-Endpoint approximation
On each subinterval for construct a rectangle with
width and height equal to , which is the function value
at the left endpoint of the subinterval. Then the area of the rectangle is
. Adding the areas of all these rectangles, we get an
approximate value for . We use the notation to denote that this is a
left-endpoint approximation of using subintervals.
or, in terms of sigma notation:
Rule: Right-Endpoint approximation
repeat steps above, but this time set rectangle heights to be equal to .
Then the area of the rectangle is . We use the notation
to denote that this is a right-endpoint approximation of using
subintervals.
or, in terms of sigma notation:
Riemann sum
Let be defined on a closed interval and let be any partition
of . Let be the width of each subinterval and for each , let be any point in. A
riemann sum is defined for as
Area under the Curve
Let be a continuous, nonnegative function on an interval , and
let be a Riemann sum for with a
regular partition . Then, the area under the curve on is
given by
If is a function defined on an interval , the definite integral of
from to
is given by
provided the limit exists. If this limit exists, the function is said to
be integrable on , or is an integrable function.
Properties of the Definite Integral
Average Value of a Function
Let be continuous over the interval . Then the average
The Fundamental Theorem of Calculus, Part 1
If is continuous over an interval , and the function is
defined by
then over
The Fundamental Theorem of Calculus, Part 2
If is continuous over the interval and is any antiderivative
of then
Subtitution for indefinite integrals
Let where is continuous over an interval, let be
continuous over the corresponding range, of , and let be an
antiderivative of . Then,
Substitution for indefinite integrals
Let where is continuous over an interval , and let
be continuous over the corresponding range of . Then,
Integrals of Exponential Functions
Integral formulas involving logs
Integral formulas resulting in inverse trig functions
Integration by parts
or, if u = f(x) and v = g(x)
Choosing and
can be chosen by selecting the first type of function appearing in the
expression which appears in the LIATE
Logarithmic functions
Inverse Trig functions
Algebraic functions
Trig functions
Exponential functions
Integration by parts for definite integrals
Let and be functions with continuous derivatives on . Then
Applications of Integrals
Finding the area between two curves
Let and be continuous functions such that over an
interval . Let denote the region bounded above by the graph of
, below the graph of , and on the left and right by lines
and , respectively. Then the area of is given by
Integral Definition of natural logarithm
for , we define the natural logarithm by
Derivative of the Natural Logarithm
For , the derivative of the natural logarithm is given by
Differential equations
A differential equation is an equation involving an unknown function and one or more of its derivatives. A solution to a differential equation
is a function that satisfies the differential equation when and
its derivatives are substituted into the equation.
the order of a differential equation is the highest order of any derivative
of the unknown function that appears in the equation.
A separable differential equation is any equation that can be written in the
form
