Calculate the arc length of the curve $\int _Lydl\:$,where $L$ - the arc $y=\sqrt{1-x^2}$

Question

Calculate the arc length of the curve $\int _Lydl\:$,where $L$ - the arc $y=\sqrt{1-x^2}$

Use the formula $$ L=\int _a^b\sqrt{1+(y')^2}dx$$

We shall find the derivative. What's next? How to find the limits of integration?

Answer 1

You have $y' = \frac {-x} {\sqrt {1-x^2}}$, so $1 + (y') ^2 = \frac 1 {1-x^2}$, therefore you have to compute $\int \limits _a ^b \frac 1 {\sqrt {1-x^2}} \Bbb d x$, which is precisely $\arcsin x \big| _a ^b = \arcsin b - \arcsin a$.

Answer 2

we will use the fact that $y = \sqrt{1-x^2}$ is the part of the unit circle in the upper half plane. therefore the arc length is $$L = \int _a^b\sqrt{1+(y')^2}dx= \cos^{-1}(a) - \cos^{-1} (b).$$