12.3.2 Use the polar equation of the lemniscate and the formula for the element of arc in polar coordinates, V(rde)2 + dr2 ds To deduce that the arc length of the lemniscate is given by do S= r ds (rde)2 + dr2 (dx)2 + (dy)2 ds And r2 cos 20 = r4 = cos2 20 ds der2 + .d0 Consider r2 = cos 20 dr Here = 2 de Differentiating on both sides: sin 20 2 2r dr 2 sin 20 de ds = d0 r2 sin 20 de dr= sin2 20 ds de r4 + r2 dr sin 20 de de r4sin2 20 ds = de Vcos2 20sin2 20 ds de ds = Integrating on both sides: de ds = r S= 12.3.3 Conclude by changing the variable of integration to r, that the total length of the lemniscate is 4dr/V1 - r4 Unlike the arcsine integrand 1/V1 - t2 which is rationalized by substituting 2v/(1+ v2) for t, the lemniscate integrand 1/V1 - t4cannot be rationalized by replacing t by any rational function.

Question

I need help with 12.3.3.please.

12.3.2 Use the polar equation of the lemniscate and the formula for the element of arc in
polar coordinates,
V(rde)2 + dr2
ds
To deduce that the arc length of the lemniscate is given by
do
S=
r
ds (rde)2 + dr2
(dx)2 + (dy)2
ds
And r2 cos 20 = r4 = cos2 20
ds der2 +
.d0
Consider r2 = cos 20
dr
Here
= 2
de
Differentiating on both sides:
sin 20 2
2r dr
2 sin 20 de
ds = d0
r2
sin 20
de
dr=
sin2 20
ds de r4 +
r2
dr
sin 20
de
de
r4sin2 20
ds =
de
Vcos2 20sin2 20
ds
de
ds =
Integrating on both sides:
de
ds =
r
S=
Expand
Transcribed Image Text

12.3.2 Use the polar equation of the lemniscate and the formula for the element of arc in polar coordinates, V(rde)2 + dr2 ds To deduce that the arc length of the lemniscate is given by do S= r ds (rde)2 + dr2 (dx)2 + (dy)2 ds And r2 cos 20 = r4 = cos2 20 ds der2 + .d0 Consider r2 = cos 20 dr Here = 2 de Differentiating on both sides: sin 20 2 2r dr 2 sin 20 de ds = d0 r2 sin 20 de dr= sin2 20 ds de r4 + r2 dr sin 20 de de r4sin2 20 ds = de Vcos2 20sin2 20 ds de ds = Integrating on both sides: de ds = r S=

12.3.3 Conclude by changing the variable of integration to r, that the total length of the
lemniscate is 4dr/V1 - r4
Unlike the arcsine integrand 1/V1 - t2 which is rationalized by substituting 2v/(1+ v2)
for t, the lemniscate integrand 1/V1 - t4cannot be rationalized by replacing t by any
rational function.
Expand
Transcribed Image Text

12.3.3 Conclude by changing the variable of integration to r, that the total length of the lemniscate is 4dr/V1 - r4 Unlike the arcsine integrand 1/V1 - t2 which is rationalized by substituting 2v/(1+ v2) for t, the lemniscate integrand 1/V1 - t4cannot be rationalized by replacing t by any rational function.

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