Riemann Sum Approximation

The definite integral of a continuous function ‘f’ over an interval [a, b] is computed as _a∫^b f(x)dx = lim Σf(x_k*)∆x, where the sum that appears on the right side is called Riemann sum.  In this formula, the interval [a, b] is divided into n subintervals of width ∆x = (b-a)/n, and x_k* denotes an arbitrary point in the k^th sub-interval  It follows that as n increases the Riemann sum will eventually be a good approximation to the integral, which we denote by writing
                  _a∫^b f(x)dx ≈ Σf(x_k*)∆x
                 _a∫^b f(x)dx ≈ ∆x[f(x₁^*) + f(x₂^*) + …….+f(x_n^*)]


Here we denote the values of ‘f’ at the endpoints of the subintervals by
y₀=f(a), y₁=f(x₁), y₂=f(x₂), ………, y_n-1=f(x_n-1), y_n=f(b) and we will denote the values of f at the midpoints of the subintervals by y_m1, y_m2, ……y_mn

Trapezoidal Approximation

The left-hand and right hand endpoint approximations are rarely used in applications; however, if we take the average of the left-hand and right hand endpoint approximations, we obtain a result, called the trapezoidal approximation, which is commonly used as,
                 _a∫^b f(x)∆x ≈  (b-a)/2n [ y₀ + 2y₁ + …..+ 2y_n-1 + y_n]

The name trapezoidal approximation can be explained by considering the case in which f(x)≥0 on [a, b], so that _a∫^b f(x)dx represents the area under f(x) over [a, b].  Geometrically, the trapezoidal approximation formula results if we approximate this area by the sum of the trapezoidal areas as shown in the figure
                                                                                       
Left end point Approximation:  The formula for evaluating left end point approximation is given by
                _a∫^b f(x)dx =  (b-a)/n [ y₀ + y₁ + ……….. + y_n-1]
Right Endpoint Approximation:  The formula for evaluating right end point approximation is given by
                       _a∫^b f(x)dx = (b-a)/n [y₁ + y₂ + …………. + y_n]
Mid-Point Approximation:  The formula for evaluating midpoint approximation is given by
              _a∫^b f(x)dx = (b-a)/n [ym₁ + ym₂ + ………..+ ymn] where m₁, m₂ ……m_n represents the mid values.
Example: Use Trapezoidal rule to approximate ₀∫^πsinx dx using n=10 sub intervals
Solution: a=0, b=π n=10 and f(x)=sinx,   (b-a)/n = π/10

₀∫^π sinx dx = (π/20)[y₀ + 2y₁ + 2y₂ + …………+2y_n-1 + y_n]
                = (π/20) [ sin(0) + 2sin(π/10) + 2sin(2π/10) + ……..sin(10π/10)
                = 1.983523538

Comparison of the Midpoint and Trapezoidal Approximations

The table below shows the comparison between midpoint and trapezoidal approximations for the function ln 2 = ₁∫²(1/x)dx with n=10 subdivisions

                    Midpoint Approximation

₁∫² (1/x)dx = (0.1)(6.928353603) = 0.692835360

                          Trapezoidal Approximation

₁∫² (1/x)dx = (0.05)(13.875428063) = 0.693771403

The value of ln 2 is rounded to nine decimal places and we have seen that midpoint approximation produces a more accurate result than the trapezoidal approximation.  Hence we can conclude that,
If f be a continuous on [a, b] and let |E_M| and |E_T| be the absolute errors that result from the midpoint and trapezoidal approximations of _a∫^bf(x)dx using n subintervals.
a)    If the graph of f is either concave up or concave down on (a, b), then |E_M|<|E_T|, which means that the error from the midpoint approximation is less than from the trapezoidal approximation.
b)    If the graph of ‘f’ is concave down on (a, b) then Tn <_a∫^b f(x)dx < M_n
c)    If the graph of ‘f’ is concave up on (a,b), then M_n < _a∫^b f(x)dx < Tn

Simpson’s Rule

Simpson’s Rule is given by

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Reference links:

• http://en.wikipedia.org/wiki/Riemann_sum
• http://en.wikipedia.org/wiki/Trapezoidal_rule
• http:/ /en.wikipedia.org/wiki/Simpson's_rule
• http://www.purplemath.com/modules/numeric.htm