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Find the exact length of the polar curve

Calculus
Tutor: None Selected Time limit: 0 Hours

r = theta ^2

0<theta<pi/4

Apr 29th, 2015

The formula is integral from 0 to pi/4 dt of sqrt(r^2(t) + (r')^2(t)).

Here r(t) = t^2, r'(t) = 2t, the integral is
int[sqrt(t^4 + 4t^2)]dt = int[t*sqrt(t^2 + 4)]dt = (1/2)int[sqrt(y + 4)]dy where y = t^2.

Further = (1/2)(2/3)[(y+4)^(3/2)] = (1/3)[(t^2 + 4)^(3/2)] with the substitution t from 0 to pi/4:

(1/3)[ ((pi/4)^2 + 4)^(3/2)  -  4^(3/2) ].

The only simplification here is 4^(3/2) = 8.

Apr 29th, 2015

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Apr 29th, 2015
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Apr 29th, 2015
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