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Double integrals as volume
Introduction to double integrals
The definition of a double integral is motivated through a hair density example.
Double integrals as iterated integrals
A description of how to convert double integrals into two single integrals.
Double integral examples
Examples of integrating double integrals over rectangles and triangles.
Examples of changing the order of integration in double integrals
Examples illustrating how to change the order of integration (or reverse the order of integration) in double integrals.
Double integrals as area
Explanation of how double integrals could be used to represent area.
Double integrals where one integration order is easier
Illustration of a region where integrating a double integral in one order is must easier than integrating in the other order.
Introduction to changing variables in double integrals
Introduction to the concepts behind a change of variables in double integrals. The transformation is illustrated with interactive graphics.
Area calculation for changing variables in double integrals
A derivation of how a mapping that changes variables in double integrals transforms area.
Double integral change of variable examples
Examples of calculating double integrals through changing variables.
The integrals of multivariable calculus
A summary of the integrals of multivariable calculus, including calculation methods and their relationship to the fundamental theorems of vector calculus.
Length, area, and volume factors
A summary of the expansion factors that arise from mappings in integrals of multivariable calculus.
Volumes by cross sections
Calculating the volume of a solid by integrating the area of its cross sections.
Volume of surfaces of revolution
Calculating the volume of created by rotating a plane region around some axis.
Illustrated example of changing variables in double integrals
An example illustrating with interactive graphics how changing variables transforms regions in the plane.
Volume calculation for changing variables in triple integrals
A derivation of how a mapping that changes variables in triple integrals transforms volume.
Introduction to triple integrals
An introduction to the definition of triple integrals as well as their formulation as iterated integrals.
Triple integral examples
Examples showing how to calculate triple integrals, including setting up the region of integration and changing the order of integration.
Introduction to a line integral of a scalar-valued function
Introduction with interactive graphics illustrating the line integral of a scalar-valued function and informally deriving the formula for calculating the integral from the parametrization of the curve.
Line integrals are independent of parametrization
Description of how the value of a line integral over a curve is independent of the parametrization of the curve.
Examples of scalar line integrals
Examples demonstrating how to calculate line integrals of scalar-valued functions.
Double integrals as volume