Calc 1 Lecture Notes
Section 4.2
Page 1 of 5
Section 4.2: Sums and Sigma Notation
Big idea: Sums are an important topic to review in preparation for the geometric concept of an
integral as the area under a curve.
Big skill: You should be able to compute a
Calc 2 Lecture Notes
Section 5.5
Page 1 of 3
Section 5.5: Projectile Motion
Big idea: Newtons Second Law is a differential equation that relates force to acceleration.
Integrating the equation once leads to an equation for the velocity, and integrating a
Calc 2 Lecture Notes
Section 5.6
Page 1 of 8
Section 5.6: Applications of Integration to Physics and Engineering
Big idea: You can integrate small pieces of work to find the total work, small changes in
momentum to find the total change in momentum, small
Calc 2 Lecture Notes
Section 6.1
Page 1 of 4
Section 6.1: Review of Integration Formulas and Techniques
Big idea: With some creative algebra, you can do a lot of new-looking integrals by
manipulating the integrand to match integral formulas from Calculus
Calc 2 Lecture Notes
Section 6.2
Page 1 of 5
Section 6.2: Integration by Parts
Big idea: Integration by parts is an integration technique that allows you to integrate the product
of two functions (under certain circumstances). Integration by parts is real
Calc 2 Lecture Notes
Section 5.7
Page 1 of 7
Section 5.7: Probability
Big idea: If you have a function that mimics a histogram of all possible outcomes for some kind
of measurement, then the definite integral of that function over a range of outcomes or
m
Calc 2 Lecture Notes
Section 6.5
Page 1 of 3
Section 6.5: Integration Tables and Computer Algebra Systems
Big idea: If the integrals weve been doing are easy enough for us to do, then a professional
mathematician must have worked out formulas for them lon
Calc 2 Lecture Notes
Section 6.4
Page 1 of 7
Section 6.4: Integration of Rational Functions Using Partial Fractions
Big idea: In this section, well examine a technique called partial fraction decomposition that is
used for evaluating the integral of a rat
Calc 2 Lecture Notes
Section 6.3
Page 1 of 8
Section 6.3: Trigonometric Techniques of Integration
Big idea: A lot of wicked-looking integrals can be computed using trigonometric identities and
substitutions.
Big skill:. You should be able to find the anti
Calc 2 Lecture Notes
Section 6.6
Page 1 of 6
Section 6.6: Improper Integrals
Big idea: An improper integral has an integrand that blows up somewhere in the interval of
integration, or has a limit of integration that is . We deal with these integrals by lo
Calc 2 Lecture Notes
Section 5.4
Page 1 of 5
Section 5.4: Arc Length and Surface Area
Big idea: The integral can be used to compute other geometric quantities besides areas under a
curve. It can be used whenever you can break up a problem into the sum of
Calc 2 Lecture Notes
Section 5.3
Page 1 of 4
Section 5.3: Volumes by Cylindrical Shells
Big idea: The integral can be used to compute other geometric quantities besides the area under
a curve. It can be used whenever you can break up a problem into the su
Calc 2 Lecture Notes
Section 5.5
Page 1 of 3
Section 5.5: Projectile Motion
Big idea: Newtons Second Law is a differential equation that relates force to acceleration.
Integrating the equation once leads to an equation for the velocity, and integrating a
Calc 1 Lecture Notes
Section 4.1
Page 1 of 8
Integration: The Other Half of Calculus
Integration Overview
1. Functionally speaking, integration is the inverse process of differentiation:
Division is the inverse process of
multiplication.
The process is ca
Calc 1 Lecture Notes
Section 4.3
Page 1 of 6
Section 4.3: Area
Big idea: The area under a curve can be numerically approximated by dividing the area up into
thin vertical rectangles of uniform width and of heights determined by the y value of the curve,
a
Calc 1 Lecture Notes
Section 4.5
Page 1 of 4
Section 4.5: The Fundamental Theorem of Calculus
Big idea: Integration and differentiation are inverse processes. That means antiderivatives can
be used to compute definite integrals.
Big skill: You should be a
Calc 1 Lecture Notes
Section 4.8
Page 1 of 2
Section 4.8: The Natural Logarithm as an Integral
Big idea: The natural logarithm can be defined as the integral of the reciprocal function. All the
rules and properties of logarithms follow from this definitio
Calc 1 Lecture Notes
Section 4.6
Page 1 of 4
Section 4.6: Integration by Substitution
Big idea: Integration by substitution is a formal way of performing the chain rule in reverse.
Big skill: You should be able to calculate integrals using substitution.
L
Calc 2 Lecture Notes
Section 5.1
Page 1 of 3
Section 5.1: Area Between Curves
Big idea: The integral can be used to compute other geometric quantities besides the area under a curve. It can
be used whenever you can break up a problem into the sum of an in
Calc 1 Lecture Notes
Section 4.7
Page 1 of 13
Section 4.7: Numerical Integration
Big idea: There are several techniques, each using different geometric shapes, for computing a
numerical approximation to a given definite integral. Some of those shapes are:
Calc 2 Lecture Notes
Section 5.2
Page 1 of 4
Section 5.2: Volume
Big idea: The integral can be used to compute other geometric quantities besides the area under
a curve. It can be used whenever you can break up a problem into the sum of an infinite number
Calc 1 Lecture Notes
Section 1.5
Page 1 of 6
Section 1.5: Limits Involving Infinity
Big Idea: Infinity () arises in limits in two ways:
1. Vertical asymptotes when the limit of a function tends toward infinity: lxim f(x) =
a
2. End behavior of functions
Calc 2 Lecture Notes
Section 7.4
Page 1 of 11
Section 7.4: Systems of First-Order Differential Equations
Big idea: Many complex real-world models require multiple differential equations to describe
each piece of the model. This results in a system of diff
Calc 2 Lecture Notes
Section 8.1
Page 1 of 12
Chapter 8: Infinite Series
The whole point of this chapter is to be able to write any given function as an infinite sum
of power function terms, with each consecutive term resulting in a better approximation t
Calc 3 Lecture Notes
Section 11.2
Page 1 of 7
Section 11.2: The Calculus of Vector-Valued Functions
Big idea: Our basic foundational notions of the calculus of scalar functions can be extended to
vector-valued functions by applying limits to all three com
Calc 3 Lecture Notes
Section 11.4
Page 1 of 7
Section 11.4: Curvature
Big idea: Curvature is a quantity that describes what it sounds like: the curviness of a curve at
any given point on the curve, or how sharply a curve is changing direction.
Big skill:
Calc 3 Lecture Notes
Section 11.6
Page 1 of 6
Section 11.6: Parametric Surfaces
Big idea: A curve is one-dimensional, and thus always can be parameterized in terms of one
independent parameter. A surface is two-dimensional, and thus requires two independe
Calc 3 Lecture Notes
Section 11.5
Page 1 of 11
Section 11.5: Tangent and Normal Vectors
Big idea: There are three mutually orthogonal vectors to any curve in 3D that can be calculated
(fairly) easily in terms of the vector-valued function that traces out
Calc 2 Lecture Notes
Section 10.6
Page 1 of 10
Section 10.6: Surfaces in Space
Big idea: Quadratic equations in three variables produce some interesting-looking surfaces that
can be understood by thinking of what the graph of the intersection of the surfa
Calc 3 Lecture Notes
Section 12.4
Page 1 of 4
Section 12.4: Tangent Planes and Linear Approximations
Big idea: The object that is tangent to a surface in three dimensions is a plane. The equation of
the tangent plane is similar in form to the equation of