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.
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 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.
The cross product
The cross product is a special way to multiply two vectors in three-dimensional space.
Parameterizing by arc length
We find a new description of curves that trivializes arc length computations.
Functions of several variables
We introduce functions that take vectors or points as inputs and output a number.
Open and Closed Sets
We generalize the notion of open and closed intervals to open and closed sets in .
We introduce differentiability for functions of several variables and find tangent planes.