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Triple integral examples
Introduction to triple integrals
An introduction to the definition of triple integrals as well as their formulation as iterated integrals.
Triple integral change of variables story
Story illustrating the process of changing variables in triple integrals.
The shadow method for determining triple integral bounds
Explanation of the shadow method for turning a triple integral into a double integral combined with a single integral in order to compute the limits of integration.
The cross section method for determining triple integral bounds
Explanation of the cross section method for turning a triple integral into a double integral combined with a single integral in order to compute the limits of integration.
Introduction to changing variables in triple integrals
Introduction to the concepts behind a change of variables in triple integrals.
Volume calculation for changing variables in triple integrals
A derivation of how a mapping that changes variables in triple integrals transforms volume.
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 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.
Area calculation for changing variables in double integrals
A derivation of how a mapping that changes variables in double integrals transforms area.
Triple integral change of variable examples
Examples of changing variables in triple integrals.
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.
Illustrated example of changing variables in double integrals
An example illustrating with interactive graphics how changing variables transforms regions in the plane.
Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus.
Multivariable chain rule examples
Examples demonstrating the chain rule for multivariable functions.
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.
Double integrals as volume
Explanation of how double integrals could be used to represent volume.
Double integrals as area
Explanation of how double integrals could be used to represent area.
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.
Triple integral examples