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Simpson's Method, Trapezoid Method and Midpoint Method for Calculating Definite Integrals

1. Numerical Integration Methods such as Simpson's Method, Trapezoid Method and Midpoint Method are used for calculating the Value of Definite Integrals.
2. The following gives the Steps for calculating Value of Definite Integral for a Function of Variable \(x\) \(f(x)\) using Simpson's Method between Interval \([a,b]\)
   1. Choose the Number of Subintervals (denoted by \(n\)) (which must be even) for Evaluation of the Integral.
   2. Calculate the Width of each Subinterval (denoted by \(h\)) as follows
      
      \(h= {\Large\frac{b-a}{n}}\)
   3. Generate \(n+1\) Points \(x_0,x_1,\cdots,x_n\) that Divide Interval \([a,b]\) into \(n\) Equal Subintervals as follows
      
      \(x_0=a,\hspace{2mm}x_1=a+h,\hspace{2mm}x_2=a+2h,\cdots,\hspace{2mm}x_n=b\)
   4. Calculate \(f(x_0),f(x_1),f(x_2),\cdots, f(x_n)\) and use it in Simpson's Formula to Approximate the Value of Definite Integral as follows
      
      \({\Large \int_{a}^{b}}f(x)\hspace{1mm}dx \approx {\Large\frac{h}{3}} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]\)
      
      Please note that in Simpson's Formula above, the value of \(f(x_0)\) and \(f(x_n)\) are present as such. Otherwise, the value of \(f(x_i)\) is Multiplied by 4 if \(i\) is Odd and is Multiplied by 2 if \(i\) is Even.
3. The following gives the Steps for calculating Value of Definite Integral for a Function of Variable \(x\) \(f(x)\) using Trapezoid Method between Interval \([a,b]\)
   1. Choose the Number of Subintervals (denoted by \(n\)) (which must be even) for Evaluation of the Integral.
   2. Calculate the Width of each Subinterval (denoted by \(h\)) as follows
      
      \(h= {\Large\frac{b-a}{n}}\)
   3. Generate \(n+1\) Points \(x_0,x_1,\cdots,x_n\) that Divide Interval \([a,b]\) into \(n\) Equal Subintervals as follows
      
      \(x_0=a,\hspace{2mm}x_1=a+h,\hspace{2mm}x_2=a+2h,\cdots,\hspace{2mm}x_n=b\)
   4. Calculate \(f(x_0),f(x_1),f(x_2),\cdots, f(x_n)\) and use it in Trapezoid Formula to Approximate the Value of Definite Integral as follows
      
      \({\Large \int_{a}^{b}}f(x)\hspace{1mm}dx \approx {\Large\frac{h}{2}} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + \cdots + 2f(x_{n-2}) + 2f(x_{n-1}) + f(x_n)]\)
      
      Please note that in Trapezoid Formula above, the value of \(f(x_0)\) and \(f(x_n)\) are present as such. Otherwise, the value of \(f(x_i)\) is Multiplied by 2.
4. The following gives the Steps for calculating Value of Definite Integral for a Function of Variable \(x\) \(f(x)\) using Midpoint Method between Interval \([a,b]\)
   1. Choose the Number of Subintervals (denoted by \(n\)) (which must be even) for Evaluation of the Integral.
   2. Calculate the Width of each Subinterval (denoted by \(h\)) as follows
      
      \(h= {\Large\frac{b-a}{n}}\)
   3. Generate \(n\) Points \(x_1,x_2,\cdots,x_n\) that are Mid Points of Subintervals as follows
      
      \(x_i=a + (i-{\Large\frac{1}{2}})h\) for \(i=1,2,3...n\)
   4. Calculate \(f(x_1),f(x_2),\cdots, f(x_n)\) and use it in Midpoint Formula to Approximate the Value of Definite Integral as follows
      
      \({\Large \int_{a}^{b}}f(x)\hspace{1mm}dx \approx h [f(x_1) + f(x_2) + \cdots + f(x_{n-1}) + f(x_n)]\)
      
      
5. In all the methods give above, the Accuracy in the Approximation of the Value of Integral Increases with the Increase in Number of Subintervals (given by \(n\)).