The following exercises deal with Fresnel integrals. 248. [T] Use Newton’s approximation of the binomial 1 − x 2 to approximate π as follows. The circle centered at ( 1 2 , 0 ) with radius 1 2 has upper semicircle y = x 1 − x . The sector of this circle bounded by the x -axis between x = 0 and x = 1 2 and by the line joining ( 1 4 , 3 4 ) , corresponds to 1 6 of the circle and has area π 24 . This sector is the union of a right triangle with height 3 4 and base 1 4 and the region below the graph between x = 0 and x = 1 4 . To find the area of this region you can write y = x 1 − x = x × (binomial expansion of 1 − x ) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate π .
The following exercises deal with Fresnel integrals. 248. [T] Use Newton’s approximation of the binomial 1 − x 2 to approximate π as follows. The circle centered at ( 1 2 , 0 ) with radius 1 2 has upper semicircle y = x 1 − x . The sector of this circle bounded by the x -axis between x = 0 and x = 1 2 and by the line joining ( 1 4 , 3 4 ) , corresponds to 1 6 of the circle and has area π 24 . This sector is the union of a right triangle with height 3 4 and base 1 4 and the region below the graph between x = 0 and x = 1 4 . To find the area of this region you can write y = x 1 − x = x × (binomial expansion of 1 − x ) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate π .
The following exercises deal with Fresnel integrals.
248. [T] Use Newton’s approximation of the binomial
1
−
x
2
to approximate
π
as follows. The circle centered at
(
1
2
,
0
)
with radius
1
2
has upper semicircle
y
=
x
1
−
x
. The sector of this circle bounded by the x-axis between x = 0 and
x
=
1
2
and by the line joining
(
1
4
,
3
4
)
, corresponds to
1
6
of the circle and has area
π
24
. This sector is the union of a right triangle with height
3
4
and base
1
4
and the region below the graph between x = 0 and
x
=
1
4
. To find the area of this region you can write
y
=
x
1
−
x
=
x
×
(binomial expansion of
1
−
x
) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate
π
.
Finite Mathematics & Its Applications (12th Edition)
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