Calculus
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Boundless Calculus
Derivatives and Integrals
Calculus Textbooks Boundless Calculus Derivatives and Integrals
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus

Section 4

Applications of Integration

Book Version 1
By Boundless
Boundless Calculus
Calculus
by Boundless
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6 concepts
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Area Between Curves

The area between the graphs of two functions is equal to the integral of a function, f(x)f(x)f(x), minus the integral of the other function, g(x)g(x)g(x): A=∫ab(f(x)−g(x))dxA = \int_a^{b} ( f(x) - g(x) ) \, dxA=∫​a​b​​(f(x)−g(x))dx.

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Volumes

Volumes of complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary.

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Average Value of a Function

The average of a function f(x)f(x)f(x) over an interval [a,b][a,b][a,b] is f¯=1b−a∫abf(x) dx\bar f = \frac{1}{b-a} \int_a^b f(x) \ dx​f​¯​​=​b−a​​1​​∫​a​b​​f(x) dx.

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Cylindrical Shells

In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.

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Work

Forces may do work on a system. Work done by a force (FFF) along a trajectory (CCC) is given as ∫CF⋅dx\int_C \mathbf{F} \cdot d\mathbf{x}∫​C​​F⋅dx.

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Volumes of Revolution

Disc and shell methods of integration can be used to find the volume of a solid produced by revolution.

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