12.3.2 Use the polar equation of the lemniscate and the formula for the element of arc inpolar coordinates,V(rde)2 + dr2dsTo deduce that the arc length of the lemniscate is given bydoS=rds (rde)2 + dr2(dx)2 + (dy)2dsAnd r2 cos 20 = r4 = cos2 20ds der2 +.d0Consider r2 = cos 20drHere= 2deDifferentiating on both sides:sin 20 22r dr2 sin 20 deds = d0r2sin 20dedr=sin2 20ds de r4 +r2drsin 20deder4sin2 20ds =deVcos2 20sin2 20dsdeds =Integrating on both sides:deds =rS= 12.3.3 Conclude by changing the variable of integration to r, that the total length of thelemniscate is 4dr/V1 - r4Unlike 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 anyrational function.

Question
Asked Nov 1, 2019

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=
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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=

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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.
help_outline

Image Transcriptionclose

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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check_circleExpert Solution
Step 1

To find the perimeter of the given lemniscate curve

Step 2

Use the arc length formula (obtained in 12,3.2)  to  write the perimeter P (taking into account the symmetry of the curve about all the axes ) , to arrive at the required expression (in r)

r^2
-cos2(theta)
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r^2 -cos2(theta)

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Step 3

Thus, the required expression for the total l...

P 4x ,with r = cos 2t,0 < t<a/4
now,2rchr=--2sin 2tdt = -2/1-r*dt
0
-rdr
4J
v-r
rdr
So,P = 4x -
as required
help_outline

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P 4x ,with r = cos 2t,0 < t<a/4 now,2rchr=--2sin 2tdt = -2/1-r*dt 0 -rdr 4J v-r rdr So,P = 4x - as required

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