Olika typer av Riemannsummor
22 december 2006
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using an Upper Riemann Sum
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using an Upper Riemann Sum
Area: 2.297682803
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using an Upper Riemann Sum
Area: 2.152965607
Area: 2.077511627
1,0
1,0
1,0
0,75
0,75
0,75
0,5
0,5
0,5
0,25
0,25
0,25
0,0
0,0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
x
−0,25
0,0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
x
−0,25
Partitions: 10
0,0
0,5
1,0
1,5
2,0
2,5
3,0
x
−0,25
Partitions: 20
Partitions: 40
f(x)
f(x)
f(x)
Figur 1: Riemannsumma där varje delrektangels höjd är funktionens max
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using a Lower Riemann Sum
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using a Lower Riemann Sum
Area: 1.669364272
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using a Lower Riemann Sum
Area: 1.838806340
Area: 1.920431996
1,0
1,0
1,0
0,75
0,75
0,75
0,5
0,5
0,5
0,25
0,25
0,25
0,0
0,0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
x
−0,25
0,0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
x
−0,25
Partitions: 10
0,0
0,5
1,0
1,5
2,0
2,5
3,0
x
−0,25
Partitions: 20
Partitions: 40
f(x)
f(x)
f(x)
Figur 2: Riemannsumma där varje delrektangels höjd är är funktionens min.
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using a Riemann Sum with Randomly Selected Points
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using a Riemann Sum with Randomly Selected Points
Area: 2.075299349
An Approximation of the Integral of
f(x) = sin(x)
on the Interval [0, Pi]
Using a Riemann Sum with Randomly Selected Points
Area: 1.965606506
Area: 2.007168829
1,0
1,0
1,0
0,75
0,75
0,75
0,5
0,5
0,5
0,25
0,25
0,25
0,0
0,0
0,0
0,5
1,0
1,5
2,0
x
−0,25
2,5
0,0
0,0
3,0
0,5
1,0
2,0
x
−0,25
Partitions: 10
1,5
3,0
0,0
0,5
1,0
1,5
2,0
2,5
x
−0,25
Partitions: 20
f(x)
2,5
Partitions: 40
f(x)
f(x)
Figur 3: Riemannsummor där höjden mäts ovanför en slumpmässigt vald punkt i delintervallen
3,0