Numerical Integration with Mathematica
Numerical Integration with Mathematica
In this notebook, I'll show you how to use Mathematica to perform numerical integration using the five methods we've discussed in class. For these examples, we'll use main integral from class, namely Integrate[Exp[-x^2],{x,0,2}]. We can find a highly accurate approximation (to 10 digits in this case), using fancy Mathematica functions, like so:
In[]:=
N[Integrate[Exp[-x^2],{x,0,2}],10]
Out[]=
0.8820813908
In[]:=
b=2;a=0;f[x_]=Exp[-x^2];
Left Hand Rule
Left Hand Rule
In[]:=
n=4.;Dx=(b-a)/n;LHR=Dx*Sum[f[a+k*Dx],{k,0,n-1}]
Out[]=
1.12604
Right Hand Rule
Right Hand Rule
In[]:=
n=4.;Dx=(b-a)/n;RHR=Dx*Sum[f[a+k*Dx],{k,1,n}]
Out[]=
0.635198
Midpoint Rule
Midpoint Rule
In[]:=
n=2.;Dx=(b-a)/n;MPR=Dx*Sum[f[a+(2k-1)Dx/2],{k,1,n}]
Out[]=
0.8842
Trapezoid Rule
Trapezoid Rule
In[]:=
n=2.;Dx=(b-a)/n;TPR=Dx*Sum[(f[a+k*Dx]+f[a+(k+1)*Dx])/2,{k,0,n-1}]
Out[]=
0.877037
Simpson's Rule
Simpson's Rule
In[]:=
n=4.;(*nmustbeeven!!*)Dx=(b-a)/n;SPR=Dx/3*Sum[(f[a+2k*Dx]+4f[a+(2k+1)*Dx]+1f[a+(2k+2)*Dx]),{k,0,n/2-1}]TPR/3+2*MPR/3
Note: Simpson's Rule with n=4 is the same as the weighted average of 1/3Trapezoid + 2/3Midpoint. But the Trapezoid and Midpoint rule use n=2 (half of four).
Out[]=
0.881812
Out[]=
0.881812