  # 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 help_outlineImage Transcriptionclose12.3.2 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= fullscreen help_outlineImage Transcriptionclose12.3.3 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. fullscreen
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)

Step 3

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

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