Tutorial
This course is built in Ximera.
A review of integration
A review of integration
We review differentiation and integration.
Areas between curves
Area between curves
We compute the area of a region between two curves using the definite integral.
Accumulated cross sections
Accumulated cross-sections
We can also use integrals to compute volume.
Accumulated shells
Accumulated shells
Some volumes of revolution are more easily computed with cylindrical shells.
Length of curves
Length of curves
We can integrate to find the length of curves.
Surface area
Surface area
We compute surface area.
Applications of integration
Phyical applications
We see several physical applications of integration.
Exponential models
Exponential and logarithmetic functions
Exponential and logarithmic functions illuminated.
The origins of a logarithm
We look at the origins of a logarithm.
Exponential models
We investigate how the exponential functions model phenomena in the real world.
Integration by parts
Integration by parts
We learn a new technique, called integration by parts, to help us solve problems involving integration.
Trigonometric integrals
Trigonometric integrals
We can substitution and trigonometric identities to antidifferentiate trigonometric functions.
Trigonometric substitution
Trigonometric substitution
We integrate by substitution with the appropriate trigonometric function.
Partial fractions
Rational functions
We can now integrate a large class of functions.
Improper integrals
Improper Integrals
We can use limits to integrate functions on unbounded domains or functions with unbounded range.
Differential equations
Differential equations
Differential equations show you relationships between rates of functions.
Numerical methods
Slope fields and Euler’s method
We describe numerical and graphical methods for understanding differential equations.
Separable differential equations
Separable differential equations
Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation.
Sequences
Sequences
A sequence is an ordered list.
Sequences as functions
Sequences as functions
A function from positive integers to the real numbers is a sequence.
Sums of sequences
Series
A series is summation of a sequence.
Integral and divergence tests
The integral test
Infinite sums can be studied using improper integrals.
The divergence test
If an infinite sum converges, then its terms must tend to zero.
Ratio and root tests
The ratio test
Some infinite series can be compared to geometric series.
The root test
Some infinite series can be compared to geometric series.
Comparison tests
The comparison test
We compare infinite series to each other using inequalities.
The limit comparison test
We compare infinite series to each other using limits.
Alternating series
The alternating series test
Alternating series have nice properties.
Approximating functions with polynomials
Approximating functions with polynomials
We can approximate smooth functions with polynomials.
Power series
Power series
Infinite series can represent functions.
Introduction to Taylor series
Introduction to Taylor series
We study Taylor and Maclaurin series.
Numbers and Taylor series
Numbers and Taylor series
Taylor series are a computational tool.
Calculus and Taylor series
Calculus and Taylor series
Power series interact nicely with other calculus concepts.
Parametric equations
Parametric equations
We discuss the basics of parametric curves.
Calculus and parametric curves
We discuss derivatives and integrals of parametric curves.
Introduction to polar coordinates
Introduction to polar coordinates
Polar coordinates are a special type of parametric curves.
Gallery of polar curves
We see a collection of polar curves.
Derivatives of polar functions
Derivatives of polar functions
We differentiate polar functions.
Integrals of polar functions
Integrals of polar functions
We integrate polar functions.
Vectors
Vectors
Vectors are lists of numbers that denote direction and magnitude.
Two and three dimensional geometry
We talk about basic geometry in higher dimensions.
Dot products
The dot product
The dot product is one way to multiply two vectors.
Cross products
The cross product
The cross product is a special way to multiply two vectors in three-dimensional space.
Lines and curves in space
Lines and curves in space
Vector-valued functions are parametrized curves.
Calculus and vector-valued functions
Calculus and vector-valued functions
With one input, and vector outputs, we work component-wise.

You can download a Certificate as a record of your successes.