CSC601 Exam. Section C
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Sample question. Use of trapezoid rule in numerical integration
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Introduction
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The trapezoid rule is an important technique for numerical integration
of mathematical functions on a given interval [a,b]. It is based on an
estimation of the area under a curve of the integrated function f(x) 
using trapezoids.
The estimation of  integral[f(x)dx]{from a to b}
is approached by first dividing the interval [a,b]
into subintervals according to the partition:

P = {a = x(1) < x(2) < x(3) < ... < x(n) = b}

For each such partition of the interval (in general the partition points x(i) 
need not be uniformly spaced),  an estimation of the integral by
the trapezoid rule is obtained.

A typical individual trapezoid has the subinterval [x(i), x(i+1)]
as its base, and the two vertical sides are f(x(i)) and f(x(i+1)).
The areas is equal to the base times the average height:

A(i) = .5*(x(i+1)-x(i))*(f(x(i))+f(x(i+1)))

The total area of trapezoids for partition P is:

T(f,P)= Sum [A(i)] {i=1, ..., n-1}            (formula 1)

Consider the case of uniform spacing, that is 
for all i: (x(i+1)-x(i)) is the same.
Let (x(i+1)-x(i)) be denoted by h.
That is, h = (b-a)/n-1 and therefore
x(i) = a + h*(i-1)

In this case (formula 1) can be re-written as follows:

T(f,P)= .5*h*Sum [f(x(i)+f(x(i+1))] {i=1, ..., n-1}


Problem Specification
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Consider the problem of calculating the integral of function f(x)=e^(-x^2)
on an interval [a,b].

(1) Outline an algorithm (using design diagrams) for a program which
receives two interval points a and b, so that a < b, and the number
of partition elements n (n is asked to be sufficiently large
in relation to the interval [a,b] inorder to guarantee the
reasonable accurracy of this integration method) and computes the
estimated value of the integral integral[f(x)dx]{from a to b}
using the trapezoid method with uniform spacing.               
                                                      [10 marks]

(2) Write a C++ implementation for this algorithm           
                                                      [10 marks]