1.3A review of integration techniques
We review common techniques to compute indefinite and definite integrals.
2.1Area 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.
3.1Accumulated cross-sections
We can also use the procedure of “Slice, Approximate, Integrate” to set up integrals
to compute volumes.
4.1What is a solid of revolution?
We define a solid of revolution and discuss how to find the volume of one in two
different ways.
4.2The washer method
We use the procedure of “Slice, Approximate, Integrate” to develop the washer
method to compute volumes of solids of revolution.
4.3The shell method
We use the procedure of “Slice, Approximate, Integrate” to develop the shell method
to compute volumes of solids of revolution.
5.1Length of curves
We can use the procedure of “Slice, Approximate, Integrate” to find the length of
curves.
6.1Surface 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.
7.1Physical applications
We apply the procedure of “Slice, Approximate, Integrate” to model physical
situations.
8.1Integration 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.
9.1Trigonometric integrals
We can use substitution and trigonometric identities to find antiderivatives of certain
types of trigonometric functions.
10.1Trigonometric substitution
We integrate by substitution with the appropriate trigonometric function.
12.1Improper Integrals
We can use limits to integrate functions on unbounded domains or functions with
unbounded range.
13.2Representing 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.
18.1The 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.
19.2Remainders 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.
20.1Remainders for alternating series
There is a nice result for approximating the remainder of convergent alternating
series.
20.2Remainders and the Integral Test
There is a nice result for approximating the remainder for series that converge by the
integral test.
23.1Absolute 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.
24.1Higher Order Polynomial Approximations
We can approximate sufficiently differentiable functions by polynomials.
30.1Slope fields and Euler’s method
We describe numerical and graphical methods for understanding differential
equations.
31.1Separable differential equations
Separable differential equations are those in which the dependent and independent
variables can be separated on opposite sides of the equation.
33.1Introduction to polar coordinates
Polar coordinates are coordinates based on an angle and a radius.
38.1The Dot Product
The dot product is an important operation between vectors that captures geometric
information.