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:
:* [[wp:Rectangle_method|rectangular]]
:**  left
:**  right
:**  midpoint
:* [[wp:Trapezoidal_rule|trapezium]]
:* [[wp:Simpson%27s_rule|Simpson's]]
:** composite

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''}}).

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:
{| class="mw-collapsible mw-collapsed"
|+ Pseudocode: Simpson's method, composite
|-
|
 '''procedure''' quad_simpson_composite(f, a, b, n)
     h := (b - a) / n
     sum1 := f(a + h/2)
     sum2 := 0

     loop on i from 1 to (n - 1)
         sum1 := sum1 + f(a + h * i + h/2)
         sum2 := sum2 + f(a + h * i)

     ''answer'' := (h / 6) * (f(a) + f(b) + 4*sum1 + 2*sum2)
|}


Demonstrate your function by showing the results for:
* {{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.
* {{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
* {{math|1=ƒ(x) = x}},    where  '''x'''  is  [0,5000],  with 5,000,000 approximations.  The exact result is  12,500,000.
* {{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'''
* [[Active object]] for integrating a function of real time.
* [[Special:PrefixIndex/Numerical integration]] for other integration methods.

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