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Question: Write the Simpsons one third rule formula in numerical methods....

Write the Simpsons one third rule formula in numerical methods.

Explanation

Solution

Now we know that the Simpson’s third rule formula in numerical methods gives us the value of definite integral for a function. The formula for Simpson’s third rule is given by ∫abf(x)dx=h3[f(x0)+f(x1)+4(f(x)1+f(x3)+....f(xn−1))+2(f(x2)+....f(xn−2))]\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{h}{3}\left[ f\left( {{x}_{0}} \right)+f\left( {{x}_{1}} \right)+4\left( f{{\left( x \right)}_{1}}+f\left( {{x}_{3}} \right)+....f\left( {{x}_{n-1}} \right) \right)+2\left( f\left( {{x}_{2}} \right)+....f\left( {{x}_{n-2}} \right) \right) \right]

Complete step-by-step answer:
Now in the field of numerical analysis, Numerical methods are methods which are formed to solve numerical problems. Numerical methods are algorithms which are used to solve particular types of problems.
Now one of the numerical methods is Simpson’s one third rule formula.
This method is widely used to find the definite integral of function. Now we know that the definite integral is nothing but the area under the curve. Hence we can say that the method is also useful to find the area under the curve in an interval.
Now in this method we first divide the interval in n parts where n is an even number.
Now we know that the interval is divided into small parts each of width ∣b−an∣\left| \dfrac{b-a}{n} \right| where a and b are the limits of the integral.
Now let a=x0,x1=a+h,x2=x1+h.....xn=ba={{x}_{0}},{{x}_{1}}=a+h,{{x}_{2}}={{x}_{1}}+h.....{{x}_{n}}=b
Hence calculate the values of x0,x1,..............xn{{x}_{0}},{{x}_{1}},..............{{x}_{n}}
Now consider y = f(x). hence find the values of y for each x0,x1,..............xn{{x}_{0}},{{x}_{1}},..............{{x}_{n}}
Now use these values in the Simpsons formula to find the integral.
The formula for Simpson’s one third rule is given by
∫abf(x)dx=h3[f(x0)+f(x1)+4(f(x)1+f(x3)+....f(xn−1))+2(f(x2)+....f(xn−2))]\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{h}{3}\left[ f\left( {{x}_{0}} \right)+f\left( {{x}_{1}} \right)+4\left( f{{\left( x \right)}_{1}}+f\left( {{x}_{3}} \right)+....f\left( {{x}_{n-1}} \right) \right)+2\left( f\left( {{x}_{2}} \right)+....f\left( {{x}_{n-2}} \right) \right) \right]

Note: Note that the rule explained above is the Simpson’s one third rule used to find definite integral. Similarly we have Simpson’s 38\dfrac{3}{8} which says the definite integral is equal to ∫abf(x)dx=3h8[f(x0)−3∑i≠3kn−1f(x1)+2∑j=1n3−1f(x3j)+f(xn)]\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{3h}{8}\left[ f\left( {{x}_{0}} \right)-3\sum\limits_{i\ne 3k}^{n-1}{f\left( {{x}_{1}} \right)}+2\sum\limits_{j=1}^{\dfrac{n}{3}-1}{f\left( {{x}_{3j}} \right)}+f\left( {{x}_{n}} \right) \right] . Also note that these both rules give us the approximate value of the integral and not the exact values.