do anyone know about simpson's rule?

Welcome to OpenStudy cute02! The Simpson's Rule is a way to approximately calculate the area under an integral.

Simpson's rule also corresponds to the 3-point Newton-Cotes quadrature rule. The method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England. Kepler used similar formulas over 100 years prior and in German the method is sometimes called Keplersche Fassregel for this reason. Simpson's rule is a staple of scientific data analysis and engineering. It is widely used, for example, by naval architects to numerically integrate hull offsets and cross-sectional areas to determine volumes and centroids of ships or lifeboats

\[\int_{a}^{b}f(x)\approx \frac{h}{3}(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]

How to Approximate Area with Simpson's Rule?

you will need 3 x values that are equally spaced and then you will need to find the functions for each x value this is 1 application of simpsons rule..

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There is an interesting example here: http://www.mathsrev.com/simpsons-rule/

For example, consider f(x)=sinx (black curve) on the interval [0,pi/2], so that f(x_0=0)=0, f(x_1=pi/4)=1/sqrt(2), and f(x_2=pi/2)=1

Sn , Simpson’s rule with n intervals Simpson’s rule uses small pieces of parabolas (that is, graphs of the kind y = ax2 + bx + c) to approximate the graphs. One can easily figure out the equation of a parabola passing through three points, say (xi−1 , f (xi−1 )), (xi , f (xi )), and (xi+1 , f (xi+1 )). and one can easily figure out a general formula for the area under a small piece of a parabola. In the end, we get b−a n = a+i x x = xi and the total approximate area under the curve is approximate area under the graph 1 x (f (x0 ) + 4f (x1 ) + 2f (x2 ) + . . . + 2f (xn−2 ) + 4f (xn−1 ) + f (xn )) .

well the solution will depend on how may sub intervals you need to use... and remember the area between pi and 2pi needs the use of absolute values since it will be negative

Approximate the integral 3 1 \[\int\limits_{1}^{3}\]x2 dx using L4 , R4 , M4 , T4 , and S4 .

so then you can do 1 application find (a + b)/2 = (1 + 3)/2 = 2 so height = 2 \[A \approx \frac{3}{3} \times[f(1) + 4 \times f(2) + f(3)] ... A \approx 1[ 1 + 4 \times 4 + 9]\]

if you use repeated applications you don't need to remember the formula.

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