In general, the length of a differentiable function can be computed using the integral:

$$\begin{aligned} \displaystyle \int_a^b \sqrt{1+f'(x)^2} \, dx \\ \end{aligned}$$

To find the length of a semi-circle, we can start with:

$$\begin{aligned} x^2 + y^2 &= r^2 \\ \end{aligned}$$

which leads to:

$$\begin{aligned} & \displaystyle \int_{-r}^r \sqrt{1+\frac{x^2}{r^2-x^2}} \, dx \\ &= \displaystyle r \int_{-\pi/2}^{\pi/2} \cos\theta \sqrt{1+\frac{r^2\sin^2\theta}{r^2-r^2\sin^2\theta}} \, d\theta \\ &= \displaystyle r \int_{-\pi/2}^{\pi/2} \cos\theta \sec \theta \, d\theta \\ &= \pi r \end{aligned}$$

$\Box$

# posted by rot13(Unafba Pune) @ 12:52 PM