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Integrating multivariable functions

Line integrals for scalar functions (videos)

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Introduction to the line integral
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Line integral example 1
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Line integral example 2 (part 1)
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Line integral example 2 (part 2)
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Line integrals for scalar functions (articles)

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

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Line integrals and vector fields
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Using a line integral to find work
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Parametrization of a reverse path
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Scalar field line integral independent of path direction
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Vector field line integrals dependent on path direction
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Path independence for line integrals
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Closed curve line integrals of conservative vector fields
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Example of closed line integral of conservative field
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Second example of line integral of conservative vector field
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Line integrals in vector fields (articles)

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

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Double integral 1
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Double integrals 2
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Double integrals 3
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Double integrals 4
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Double integrals 5
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Double integrals 6
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Double integrals (articles)

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Double integrals
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Double integrals over non-rectangular regions
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Double integrals beyond volume
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Polar coordinates
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Double integrals in polar coordinates
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Triple integrals (videos)

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Triple integrals 1
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Triple integrals 2
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Triple integrals 3
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Triple integrals (articles)

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Triple integrals
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Triple integrals in cylindrical coordinates
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Triple integrals in spherical coordinates
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Surface integral preliminaries (videos)

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Parametrizing a surface, part 1
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Determining a position vector-valued function for a parametrization of two parameters
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Partial derivatives of vector-valued functions
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Surface integrals (videos)

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Introduction to the surface integral
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Example of calculating a surface integral part 1
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Example of calculating a surface integral part 2
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Example of calculating a surface integral part 3
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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
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Surface integral ex3 part 2
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Surface integral ex3 part 3
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Surface integral ex3 part 4
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Surface integrals (articles)

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Surface area integrals
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Surface area example
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Surface integrals
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Surface integral example
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Flux in 3D (videos)

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Conceptual understanding of flux
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Constructing a unit normal vector
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Vector representation of a surface integral
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Flux in 3D (articles)

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Unit normal vector of a surface
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Flux in three dimensions
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Flux in 3D example
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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.