hi! Could anyone help me with this, What is Trapezoidal rule and Simpson's rule? thank you.

Question

badhi · Answer

These are techniques to find the definite integral of a function in numerical method http://en.wikipedia.org/wiki/Simpson's_rule http://en.wikipedia.org/wiki/Trapezoidal_rule

anonymous · Answer

thank you.

badhi · Answer

you're welcome

anonymous · Answer

i have another question. can you show me how it derive ?

badhi · Answer

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Deriving the trapezoidal rule is easy. It assumes that the area under curve between x=a and x=b is a area of the trapezoid in the diagram 
thus the area A,
$$A=\frac{f(a)+f(b)}{2}(b-a)$$

anonymous · Answer

thank you. can you show me the steps by steps process to derive it. :)

badhi · Answer

I've given you the idea. you should be able to create your own steps

badhi · Answer

proof for simpson's rule -- http://planetmath.org/ProofOfSimpsonsRule.html

anonymous · Answer

thank you! how about the trapezoidal ? :)

anonymous · Answer

hi.. Could you help me again. 

I know that the Trapezoidal Rule is a technique for approximating the definite integral $$\int\limits_{a}^{b?} f(x) dx$$ with the use of trapezoid and calculating its are It follows that
$$\int\limits_{a}^{b} f(x) = \frac{ a-b }{ n } [ f(x0) + f(x1) ... + f(xn) ]$$.

how about the truncation error of the Trapezoidal Rule?  can you gave me the definition and formula. thank you. :)

anonymous · Answer

http://en.wikipedia.org/wiki/Trapezoidal_rule#Error_analysis As far as I understand it, the error only really exists if N is relatively, arbitrarily finite (large N means some percent error. As N-> infinity, error -> 0. )