Tutorial

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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.