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Task/Numerical-integration/00-TASK.txt

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+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)
+                                            &nbsp;
+     ''answer'' := (h / 6) * (f(a) + f(b) + 4*sum1 + 2*sum2)
+|}
+
+
+Demonstrate your function by showing the results for:
+* &nbsp; {{math|1=ƒ(x) = x<sup>3</sup>}}, &nbsp; &nbsp; &nbsp;        where &nbsp; '''x''' &nbsp; is &nbsp; &nbsp; [0,1], &nbsp; &nbsp; &nbsp; with &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 100 approximations. &nbsp; The exact result is &nbsp; &nbsp; 0.25 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (or 1/4)
+* &nbsp; {{math|1=ƒ(x) = 1/x}},           &nbsp; &nbsp;               where &nbsp; '''x''' &nbsp; is &nbsp; [1,100],      &nbsp; &nbsp; with &nbsp; &nbsp; &nbsp; &nbsp;1,000 approximations. &nbsp; The exact result is &nbsp; &nbsp; 4.605170<sup>+</sup> &nbsp; &nbsp; (natural log of 100)
+* &nbsp; {{math|1=ƒ(x) = x}},             &nbsp; &nbsp; &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
+* &nbsp; {{math|1=ƒ(x) = x}},             &nbsp; &nbsp; &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
+
+<br/>
+;See also:
+* &nbsp; [[Active object]] for integrating a function of real time.
+* &nbsp; [[Special:PrefixIndex/Numerical integration]] for other integration methods.
+
+<br/>
+