Newton's Pi Approximation
Newton's Pi Approximation
This Demonstration gives Newton's approximation of based on calculating the area of a semicircle using an integral.
π
The area under the arc (light blue) is equal to the area of the sector minus the area of triangle (light green), that is, -.
α
AD
ADC
BCD
π
24
3
32
On the other hand, the semicircle has equation , so . Set =+++... and =+2.
y=
x-
=2
x
x
1-x
α=----…
1/4
∫
0
x
1-x
dx=1
12
1
160
1
3584
1
36864
S
1
1
160
1
3584
1
36864
S
2
3
3
4
Therefore, .
π=24+-=+2-24=-24
3
32
1
12
S
1
3
3
4
S
1
S
2
S
1