Numerical integration: Difference between revisions

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{{task|Arithmetic operations}}
{{task|Arithmetic operations}}


Write functions to calculate the definite integral of a function &nbsp; &nbsp; <big><big> <span style="font-family: serif">''ƒ(x)''</span> </big></big> &nbsp; &nbsp; using &nbsp; ''all'' &nbsp; five of the following methods:
Write functions to calculate the definite integral of a function <big><big> {{math|1=''ƒ(x)''}} </big></big> using ''all'' five of the following methods:
::* &nbsp; [[wp:Rectangle_method|rectangular]]
:* [[wp:Rectangle_method|rectangular]]
::::* &nbsp; left
:** left
::::* &nbsp; right
:** right
::::* &nbsp; midpoint
:** midpoint
::* &nbsp; [[wp:Trapezoidal_rule|trapezium]]
:* [[wp:Trapezoidal_rule|trapezium]]
::* &nbsp; [[wp:Simpson%27s_rule|Simpson's]]
:* [[wp:Simpson%27s_rule|Simpson's]] (composite)


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Your functions should take in the upper and lower bounds &nbsp; (<span style="font-family: serif">''a''</span> &nbsp; and &nbsp; <span style="font-family: serif">''b''</span>), &nbsp; and the number of approximations to make in that range &nbsp; (<span style="font-family: serif">''n''</span>).


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Your functions should take in the upper and lower bounds ({{math|''a''}} and {{math|''b''}}), and the number of approximations to make in that range ({{math|''n''}}).
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Assume that your example already has a function that gives values for &nbsp; &nbsp; <big> <span style="font-family: serif">''ƒ(x)''</span>. </big>

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Assume that your example already has a function that gives values for <big> {{math|1=''ƒ(x)''}} </big>.


Simpson's method is defined by the following pseudo-code:
Simpson's method is defined by the following pseudo-code:
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Demonstrate your function by showing the results for:
Demonstrate your function by showing the results for:
* <big> ƒ(x) = x<sup>3</sup>, </big> &nbsp; where &nbsp; &nbsp; '''x''' &nbsp; &nbsp; is &nbsp; [0,1], &nbsp; with 100 approximations. &nbsp; The exact result is &nbsp; 1/4, &nbsp; or &nbsp; 0.25.
* {{math|1=ƒ(x) = x<sup>3</sup>}}, where '''x''' is [0,1], with 100 approximations. The exact result is 1/4, or 0.25.
* <big> ƒ(x) = 1/x, </big> &nbsp; where &nbsp; '''x''' &nbsp; is &nbsp; [1,100], &nbsp; with 1,000 approximations. &nbsp; The exact result is the natural log of 100, &nbsp; or about &nbsp; 4.605170
* {{math|1=ƒ(x) = 1/x}}, where '''x''' is [1,100], with 1,000 approximations. The exact result is the natural log of 100, or about 4.605170
* <big> ƒ(x) = x, </big> &nbsp; &nbsp; where &nbsp; '''x''' &nbsp; is &nbsp; [0,5000], &nbsp; with 5,000,000 approximations. &nbsp; The exact result is &nbsp; 12,500,000.
* {{math|1=ƒ(x) = x}}, where '''x''' is [0,5000], with 5,000,000 approximations. The exact result is 12,500,000.
* <big> ƒ(x) = x, </big> &nbsp; &nbsp; where &nbsp; '''x''' &nbsp; is &nbsp; [0,6000], &nbsp; with 6,000,000 approximations. &nbsp; The exact result is &nbsp; 18,000,000.
* {{math|1=ƒ(x) = x}}, where '''x''' is [0,6000], with 6,000,000 approximations. The exact result is 18,000,000.


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'''See also'''
'''See also'''
* [[Active object]] for integrating a function of real time.
* [[Active object]] for integrating a function of real time.
* [[Numerical integration/Gauss-Legendre Quadrature]] for another integration method.
* [[Special:PrefixIndex/Numerical integration]] for other integration methods.

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=={{header|ActionScript}}==
=={{header|ActionScript}}==
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