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Double integrals as area
Area calculation for changing variables in double integrals
A derivation of how a mapping that changes variables in double integrals transforms area.
Double integrals as volume
Explanation of how double integrals could be used to represent 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 integral change of variable examples
Examples of calculating double integrals through changing variables.
Double integral examples
Examples of integrating double integrals over rectangles and triangles.
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.
Introduction to double integrals
The definition of a double integral is motivated through a hair density example.
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.
Area and definite integrals
Integrating to find the area under a curve or the area between two curves.
Double integrals as iterated integrals
A description of how to convert double integrals into two single integrals.
Length, area, and volume factors
A summary of the expansion factors that arise from mappings in integrals of multivariable calculus.
Using Green's theorem to find area
A trick to use Green's theorem to calculate the area of a region
Illustrated example of changing variables in double integrals
An example illustrating with interactive graphics how changing variables transforms regions in the plane.
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.
Triple integral examples
Examples showing how to calculate triple integrals, including setting up the region of integration and changing the order 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.
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.
Calculating the area under a curve using Riemann sums
Approximating the area under the graph of a positive function as sum of the areas of rectangles.
Introduction to triple integrals
An introduction to the definition of triple integrals as well as their formulation as iterated integrals.
Introduction to changing variables in triple integrals
Introduction to the concepts behind a change of variables in triple integrals.
Double integrals as area