Question

Find the arc length function for the curve with starting point (0,1).

Answer 1

What curve?

Answer 2

\[y=\sin^{-1} x + \sqrt{1-x^2}\]

Answer 3

\[s(x)=\int_{0}^{x} \sqrt{1+\left ( \frac{1}{\sqrt{1-x^2}}-\frac{x}{\sqrt{1-x^2}}\right )^2}\ dx
\\
=\int_{0}^{x} \sqrt{1+\left ( \frac{1-x}{\sqrt{1-x^2}}\right )^2}\ dx\\
=\int_{0}^{x} \sqrt{1+\frac{(1-x)^2}{1-x^2}}\ dx\\
=\int_0^{x} \sqrt{1+\frac{1-x}{1+x}}\ dx\\
=\int_0^{x} \sqrt{\frac{2}{1+x}}\ dx\]
to be continued....

Answer 4

\[s(x)=\sqrt{2}\int_0^x (1+x)^{-1/2}\ dx\\
=2\sqrt{2} \sqrt{1+x}|_0^x\\
=2\sqrt2(\sqrt{1+x}-1)\]
Whew!