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...

I need help with 12.3.3.please.

Extracted 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=


Extracted 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.