MATH 166 SCW, 7.7 - Example 1: Using Mathematica to Compute Riemann Sums

Here are some commands you can define in Mathematica to estimate the definite integral
b
∫
a
f(x)x
:
In[]:=
LeftSum[f_,a_,b_,n_]:=
n
∑
i=1
f[a+(i-1)*(b-a)/n]*(b-a)/n
In[]:=
RightSum[f_,a_,b_,n_]:=
n
∑
i=1
f[a+i*(b-a)/n]*(b-a)/n
In[]:=
MidpointSum[f_,a_,b_,n_]:=
n
∑
i=1
f[a+(2*i-1)*(b-a)/(2*n)]*(b-a)/n
In[]:=
Trapezoid[f_,a_,b_,n_]:=
n
∑
i=1
(1/2)(f[a+(i-1)*(b-a)/n]+f[a+i*(b-a)/n])*(b-a)/n
Herearesomeapproximationsofthedefiniteintegral
1
∫
0
Sin[x]x:
RightSum[Sin,0,1,4]
Out[]=
1
4
Sin
1
4
+
1
4
Sin
1
2
+
1
4
Sin
3
4
+
Sin[1]
4
N[%]
Out[]=
0.562485
RightSum[Sin,0,1,10]
Out[]=
1
10
Sin
1
10
+
1
10
Sin
1
5
+
1
10
Sin
3
10
+
1
10
Sin
2
5
+
1
10
Sin
1
2
+
1
10
Sin
3
5
+
1
10
Sin
7
10
+
1
10
Sin
4
5
+
1
10
Sin
9
10
+
Sin[1]
10
N[%]
Out[]=
0.501388
LeftSum[Sin,0,1,10]
Out[]=
1
10
Sin
1
10
+
1
10
Sin
1
5
+
1
10
Sin
3
10
+
1
10
Sin
2
5
+
1
10
Sin
1
2
+
1
10
Sin
3
5
+
1
10
Sin
7
10
+
1
10
Sin
4
5
+
1
10
Sin
9
10

N[%]
Out[]=
0.417241
MidpointSum[Sin,0,1,10]
Out[]=
1
10
Sin
1
20
+
1
10
Sin
3
20
+
1
10
Sin
1
4
+
1
10
Sin
7
20
+
1
10
Sin
9
20
+
1
10
Sin
11
20
+
1
10
Sin
13
20
+
1
10
Sin
3
4
+
1
10
Sin
17
20
+
1
10
Sin
19
20

N[%]
Out[]=
0.459889
Trapezoid[Sin,0,1,10]
Out[]=
1
20
Sin
1
10
+
1
20
Sin
1
10
+Sin
1
5
+
1
20
Sin
1
5
+Sin
3
10
+
1
20
Sin
3
10
+Sin
2
5
+
1
20
Sin
2
5
+Sin
1
2
+
1
20
Sin
1
2
+Sin
3
5
+
1
20
Sin
3
5
+Sin
7
10
+
1
20
Sin
7
10
+Sin
4
5
+
1
20
Sin
4
5
+Sin
9
10
+
1
20
Sin
9
10
+Sin[1]
N[%]
Out[]=
0.459315
Toestimateanintegralsuchas
2
∫
-1
x^2x,definethefunctionf[x_]:=x^2andusetheabovecommandsthisway:
In[]:=
f[x_]:=x^2
N[LeftSum[f,-1,2,100]]
Out[]=
2.95545
Here is the actual answer:
2
∫
-1
x^2x
Out[]=
3