Question

can someone help me with an arc length problem
Let |a|≤1. Compute the distance between (a,√(1-a^2 )) and (1, 0) along the graph of y=√(1-x^2 )

Answer 1

trig substitution, x = sin theta, and it all comes out very simply.

Answer 2

Here's an approach that doesn't involve calculus.

For \(0\le t\le\pi\), \(\sin t=\sqrt{1-a^2}\) and \(\cos t=a\). The arc length \(L\) of an arc that subtends a central angle with measure \(t\) radians can be determined by comparing the ratio of \(L\) to \(t\) to the ratio of the total circumference of the circle to one revolution (i.e. \(2\pi\) radians):
\[\frac{L}{t\text{ rad}}=\frac{2\pi}{2\pi\text{ rad}}~~\implies~~\frac{L}{\arccos a}=1\]