Question

Determine the length of the curve 𝑥 = 𝑦^2 /2 for 0 ≤ 𝑥 ≤ 1/2 . Assume 𝑦 positive.

Answer 1

length of the curve:

$L=\int\sqrt{1+(y')^2}dx$

$y=\sqrt{2x}$

$y'=1/(\sqrt{2x})$

$L=\int^{1/2}_0\sqrt{1+1/(2x)}dx=(x\sqrt{1+1/(2x)}+\frac{ln(\sqrt{2+1/x}+\sqrt2)-ln(\sqrt{2+1/x}-\sqrt2))}{4}|^{1/2}_0=$

$=1/\sqrt 2+\frac{ln(\sqrt2+1)-ln(\sqrt2-1)}{4}$