A review of integration techniques
We review common techniques to compute indefinite and definite integrals.
Area between curves
We introduce the procedure of “Slice, Approximate, Integrate” and use it study the area of a region between two curves using the definite integral.
We can also use the procedure of “Slice, Approximate, Integrate” to set up integrals to compute volumes.
What is a solid of revolution?
We define a solid of revolution and discuss how to find the volume of one in two different ways.
The washer method
We use the procedure of “Slice, Approximate, Integrate” to develop the washer method to compute volumes of solids of revolution.
The shell method
We use the procedure of “Slice, Approximate, Integrate” to develop the shell method to compute volumes of solids of revolution.
Length of curves
We can use the procedure of “Slice, Approximate, Integrate” to find the length of curves.
Surface areas of revolution
We compute surface area of a frustrum then use the method of “Slice, Approximate, Integrate” to find areas of surface areas of revolution.
We apply the procedure of “Slice, Approximate, Integrate” to model physical situations.
Integration by parts
We learn a new technique, called integration by parts, to help find antiderivatives of certain types of products by reexamining the product rule for differentiation.
We can use substitution and trigonometric identities to find antiderivatives of certain types of trigonometric functions.
We can use limits to integrate functions on unbounded domains or functions with unbounded range.
Representing sequences visually
We can graph the terms of a sequence and find functions of a real variable that coincide with sequences on their common domains.
The alternating series test
Alternating series are series whose terms alternate in sign between positive and negative. There is a powerful convergence test for alternating series.
Remainders: First Principles
We can approximate the value of convergent series using the sequence of partial sums.
Remainders for Geometric and Telescoping Series
For a convergent geometric series or telescoping series, we can find the exact error made when approximating the infinite series using the sequence of partial sums.
Remainders for alternating series
There is a nice result for approximating the remainder of convergent alternating series.
Remainders and the Integral Test
There is a nice result for approximating the remainder for series that converge by the integral test.
Absolute and Conditional Convergence
The basic question we wish to answer about a series is whether or not the series converges. If a series has both positive and negative terms, we can refine this question and ask whether or not the series converges when all terms are replaced by their absolute values. This is the distinction between absolute and conditional convergence, which we explore in this section.
Slope fields and Euler’s method
We describe numerical and graphical methods for understanding 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.
The Dot Product
The dot product is an important operation between vectors that captures geometric information.
Projections and orthogonal decomposition
Projections tell us how much of one vector lies in the direction of another and are important in physical applications.