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