Answer
399.9k+ views
Hint: 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 $\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{h}{3}\left[ f\left( {{x}_{0}} \right)+f\left( {{x}_{1}} \right)+4\left( f{{\left( x \right)}_{1}}+f\left( {{x}_{3}} \right)+....f\left( {{x}_{n-1}} \right) \right)+2\left( f\left( {{x}_{2}} \right)+....f\left( {{x}_{n-2}} \right) \right) \right]$
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 $\left| \dfrac{b-a}{n} \right|$ where a and b are the limits of the integral.
Now let $a={{x}_{0}},{{x}_{1}}=a+h,{{x}_{2}}={{x}_{1}}+h.....{{x}_{n}}=b$
Hence calculate the values of ${{x}_{0}},{{x}_{1}},..............{{x}_{n}}$
Now consider y = f(x). hence find the values of y for each ${{x}_{0}},{{x}_{1}},..............{{x}_{n}}$
Now use these values in the Simpsons formula to find the integral.
The formula for Simpson’s one third rule is given by
$\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{h}{3}\left[ f\left( {{x}_{0}} \right)+f\left( {{x}_{1}} \right)+4\left( f{{\left( x \right)}_{1}}+f\left( {{x}_{3}} \right)+....f\left( {{x}_{n-1}} \right) \right)+2\left( f\left( {{x}_{2}} \right)+....f\left( {{x}_{n-2}} \right) \right) \right]$
Note: Note that the rule explained above is the Simpson’s one third rule used to find definite integral. Similarly we have Simpson’s $\dfrac{3}{8}$ which says the definite integral is equal to $\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{3h}{8}\left[ f\left( {{x}_{0}} \right)-3\sum\limits_{i\ne 3k}^{n-1}{f\left( {{x}_{1}} \right)}+2\sum\limits_{j=1}^{\dfrac{n}{3}-1}{f\left( {{x}_{3j}} \right)}+f\left( {{x}_{n}} \right) \right]$ . Also note that these both rules give us the approximate value of the integral and not the exact values.
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 $\left| \dfrac{b-a}{n} \right|$ where a and b are the limits of the integral.
Now let $a={{x}_{0}},{{x}_{1}}=a+h,{{x}_{2}}={{x}_{1}}+h.....{{x}_{n}}=b$
Hence calculate the values of ${{x}_{0}},{{x}_{1}},..............{{x}_{n}}$
Now consider y = f(x). hence find the values of y for each ${{x}_{0}},{{x}_{1}},..............{{x}_{n}}$
Now use these values in the Simpsons formula to find the integral.
The formula for Simpson’s one third rule is given by
$\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{h}{3}\left[ f\left( {{x}_{0}} \right)+f\left( {{x}_{1}} \right)+4\left( f{{\left( x \right)}_{1}}+f\left( {{x}_{3}} \right)+....f\left( {{x}_{n-1}} \right) \right)+2\left( f\left( {{x}_{2}} \right)+....f\left( {{x}_{n-2}} \right) \right) \right]$
Note: Note that the rule explained above is the Simpson’s one third rule used to find definite integral. Similarly we have Simpson’s $\dfrac{3}{8}$ which says the definite integral is equal to $\int\limits_{a}^{b}{f\left( x \right)dx}=\dfrac{3h}{8}\left[ f\left( {{x}_{0}} \right)-3\sum\limits_{i\ne 3k}^{n-1}{f\left( {{x}_{1}} \right)}+2\sum\limits_{j=1}^{\dfrac{n}{3}-1}{f\left( {{x}_{3j}} \right)}+f\left( {{x}_{n}} \right) \right]$ . Also note that these both rules give us the approximate value of the integral and not the exact values.
Recently Updated Pages
The deliquescent among the following is ACaCl2 BFeSO47H2O class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The d electron configurations of Cr2 + Mn2 + Fe2 + class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The degree of ionization of a 01M bromoacetic acid class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The degree of Hydrolysis of CH3COONH4 is independent class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The degree of hydrolysis for a salt of strong acid class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The degree of hydrolysis for a salt of strong acid class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the full form of AD a After death b Anno domini class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name 10 Living and Non living things class 9 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)