Numerical Integration Methods Part 2

Within our lab, we plan to explore, technically 6, different methods of numerical integration, though a couple of the methods are very similar or even just different variations of the same method.  While Luke’s post will tell you all about why numerical integration is important, and why you would want to have numerical ways to compute integrals, I will provide some of the important definitions to help you remember the specifics about the different types of numerical methods of integration and the different advantages and disadvantages each method has. I’ll be covering the left-hand sum, right-hand sum, and Simpson’s rule.

The left and right-hand sums are actually variations of the Riemann sum method. All methods approximate the curve by finding the area of rectangles that cover a similar area as the curve.  As Kenny will elaborate on in his definition of the Riemann sum, these methods are calculated by dividing the interval over which a given function is to be integrated, into subintervals which will serve as the base of the rectangles which will be formed. The height of the rectangles is determined by picking a point on the function. Then the areas of each rectangle is found and added together. The sum of the areas is the approximation of the integral.

Left-Hand Sum

The distinguishing factor for a specific type of sum is the point used from which the height is determined. In the left-hand sum, the left-hand end-point of the sub interval is used as to determine the height. This means the length of the rectangle is found by extending the point a the left-hand side of the subinterval to the function. A picture I found on Wikipedia helps illustrate this concept:

File:LeftRiemann2.svg

As can be seen in the illustration, for monotonically increasing functions, the left-hand sum approximation is an underestimate of the integral. For monotonically decreasing functions, this method provides and over-estimation.

The formula for this method is as follows:

$L_n = \sum_{i=0}^{n-1}f(x)d$

Where f(x) is the function we’re integrating and d is the width of the subinterval.

Right-Hand Sum

Conversely, the right-hand sum method, uses the right-hand point of the subintervals. And can be pictured as:

File:RightRiemann2.svg

With a formula of: $R_n = \sum_{i=1}^{n}f(x)d$

Where f(x) is the function we’re integrating and d is the width of the subinterval.

Wikipedia actaully had a nice summary if you’d like more explanation: http://en.wikipedia.org/wiki/Riemann_sum#Left_sum

Simpson’s rule

SImpson’s rule allows us to compute an integral using quadratic polynomials. This method, like the other methods, separates the area that is to be integrated into subintervals, but differs in the sense that it finds the area of parabolas that encompass the subinterval, by using quadratic polynomials which approximate the function.

Simpson’s rule is formally defined as: $\frac{(b-a)}{6}[f(a)+4f(\frac{a+b}{2})+f(b)]$ and it is very accurate when calculating integrals of polynomials to a cubic degree.

Wolrfram MathWorld has a very succinct description of Simpsons rule which can be found at: http://mathworld.wolfram.com/SimpsonsRule.html.

Also, an interesting fact from Wikipedia about Simpson’s rule, this rule is widely used by naval architects to numerically integrate hull offsets and cross-sectional areas to determine volumes and centroids of ships or lifeboats.

http://en.wikipedia.org/wiki/Simpson%27s_Rule

 

 

 

2 thoughts on “Numerical Integration Methods Part 2

  1. Samuel Coskey

    Interesting that Simpson’s Rule is truly used in practical situations!

    I have the same question as in your group’s previous post: The formula you give for $L_n$, strictly speaking, doesn’t communicate anything because we don’t know what is $x$ and what is $h$. The picture is indeed very clear, but please be as precise as possible!

  2. Pingback: Numerical Integration: Summary and Conclusions | MATH 287

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