I'm using Sage to calculate a curve arc length.
Basically, the curve is given by this equation:
In sage, I'm calculating it's arc length by using this formula:
var('t')
integral(sqrt((derivative(t - sin(t)))^2 + (derivative(1 - cos(t)))^2 + (derivative(4 * cos(t/2)))^2), t, 0, 8*pi);
The formula is from here: http://www.mathwords.com/a/arc_length_of_a_curve.htm
Basically, it should return
32 * sqrt(2)
but it always returns 0.
Could anyone tell me why? I probably did something wrong with the syntax, but can't figure out where.
Probably Sage did not keep track of branches, and used a discontinuous anti-derivative. Maple, also, has a discontinuous anti-derivative. But Maple manages to take the jumps into account and gets the right answer. $$ \begin{align} &\int 2 \sqrt{2} \mathrm{sgn} \Biggl(\operatorname{sin} \biggl(\frac{t}{2}\biggr)\Biggr) \operatorname{sin} \biggl(\frac{t}{2}\biggr) d t = -4\sqrt{2} \mathrm{sgn} \Biggl(\operatorname{sin} \biggl(\frac{t}{2}\biggr)\Biggr) \operatorname{cos} \biggl(\frac{t}{2}\biggr) \\ &\int_{0}^{8 \pi} 2 \sqrt{2} \mathrm{sgn} \Biggl(\operatorname{sin} \biggl(\frac{t}{2}\biggr)\Biggr) \operatorname{sin} \biggl(\frac{t}{2}\biggr) d t = 32 \sqrt{2} \end{align} $$