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Unit: Integrating multivariable functions

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Line integrals for scalar functions

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Introduction to the line integral
(Opens a modal)
Line integral example 1
(Opens a modal)
Line integral example 2 (part 1)
(Opens a modal)
Line integral example 2 (part 2)
(Opens a modal)
Practice
Line integrals of scalar functionsGet 3 of 4 questions to level up!

Line integrals for scalar functions (articles)

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Arc length of function graphs, introduction
(Opens a modal)
Arc length of function graphs, examples
(Opens a modal)
Arc length of parametric curves
(Opens a modal)
Notation for integrating along a curve
(Opens a modal)
Line integrals in a scalar field
(Opens a modal)

Line integrals in vector fields

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Line integrals and vector fields
(Opens a modal)
Using a line integral to find work
(Opens a modal)
Parametrization of a reverse path
(Opens a modal)
Scalar field line integral independent of path direction
(Opens a modal)
Vector field line integrals dependent on path direction
(Opens a modal)
Path independence for line integrals
(Opens a modal)
Closed curve line integrals of conservative vector fields
(Opens a modal)
Example of closed line integral of conservative field
(Opens a modal)
Second example of line integral of conservative vector field
(Opens a modal)
Practice
Line integrals in vector fieldsGet 3 of 4 questions to level up!
Line integrals in conservative vector fieldsGet 3 of 4 questions to level up!
Distinguishing conservative vector fieldsGet 3 of 4 questions to level up!
Potential functionsGet 3 of 4 questions to level up!

Line integrals in vector fields (articles)

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Line integrals in a vector field
(Opens a modal)
Fundamental theorem of line integrals
(Opens a modal)
Conservative vector fields
(Opens a modal)
Flux in two dimensions
(Opens a modal)
Constructing a unit normal vector to curve
(Opens a modal)

Quiz 1

Level up on the above skills and collect up to 500 Mastery points

Double integrals

Learn
Double integral 1
(Opens a modal)
Double integrals 2
(Opens a modal)
Double integrals 3
(Opens a modal)
Double integrals 4
(Opens a modal)
Double integrals 5
(Opens a modal)
Double integrals 6
(Opens a modal)
Practice
Iterated integralsGet 3 of 4 questions to level up!
Double integrals with variable boundsGet 3 of 4 questions to level up!
Finding bounds of regionsGet 3 of 4 questions to level up!
Switching bounds on double integralsGet 3 of 4 questions to level up!

Double integrals (articles)

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Double integrals
(Opens a modal)
Double integrals over non-rectangular regions
(Opens a modal)
Double integrals beyond volume
(Opens a modal)
Polar coordinates
(Opens a modal)
Double integrals in polar coordinates
(Opens a modal)

Quiz 2

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Triple integrals

Learn
Triple integrals 1
(Opens a modal)
Triple integrals 2
(Opens a modal)
Triple integrals 3
(Opens a modal)
Triple integrals
(Opens a modal)
Practice
Triple integralsGet 3 of 4 questions to level up!

Change of variables

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No videos or articles available in this lesson
Practice
Change of variables: BoundGet 3 of 4 questions to level up!
Change of variables: FactorGet 3 of 4 questions to level up!

Quiz 3

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Polar, spherical, and cylindrical coordinates

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Triple integrals in cylindrical coordinates
(Opens a modal)
Triple integrals in spherical coordinates
(Opens a modal)
Practice
Double integrals in polarGet 3 of 4 questions to level up!
Integrals in spherical and cylindrical coordinatesGet 3 of 4 questions to level up!

Surface integral preliminaries

Learn
Parametrizing a surface, part 1
(Opens a modal)
Determining a position vector-valued function for a parametrization of two parameters
(Opens a modal)
Partial derivatives of vector-valued functions
(Opens a modal)

Surface integrals

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Introduction to the surface integral
(Opens a modal)
Example of calculating a surface integral part 1
(Opens a modal)
Example of calculating a surface integral part 2
(Opens a modal)
Example of calculating a surface integral part 3
(Opens a modal)
Surface integral example, part 1
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Surface integral example part 2
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Surface integral example part 3: The home stretch
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Surface integral ex2 part 1
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Surface integral ex2 part 2
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Surface integral ex3 part 1
(Opens a modal)
Surface integral ex3 part 2
(Opens a modal)
Surface integral ex3 part 3
(Opens a modal)
Surface integral ex3 part 4
(Opens a modal)
Practice
Find area elementsGet 3 of 4 questions to level up!
Surface integrals to find surface areaGet 3 of 4 questions to level up!

Surface integrals (articles)

Learn
Surface area integrals
(Opens a modal)
Surface area example
(Opens a modal)
Surface integrals
(Opens a modal)
Surface integral example
(Opens a modal)

Quiz 4

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Flux in 3D

Learn
Conceptual understanding of flux
(Opens a modal)
Constructing a unit normal vector
(Opens a modal)
Vector representation of a surface integral
(Opens a modal)

Flux in 3D (articles)

Learn
Unit normal vector of a surface
(Opens a modal)
Flux in three dimensions
(Opens a modal)
Flux in 3D example
(Opens a modal)
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Unit test

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About this unit

There are many ways to extend the idea of integration to multiple dimensions: Line integrals, double integrals, triple integrals, surface integrals, etc. Each one lets you add infinitely many infinitely small values, where those values might come from points on a curve, points in an area, points on a surface, etc. These are all very powerful tools, relevant to almost all real-world applications of calculus. In particular, they are an invaluable tool in physics.