29.33. Calculate the length of a circle of radius $R$ using a definite integral.
29.33. The length of the circumference is equal to twice the length of the semicircle $y=\sqrt{R^{2}-x^{2}}$. It is clear that $y^{\prime}=\frac{-x}{\sqrt{R^{2}-x^{2}}}$, so the length of the semicircle is

$$
\lim _{\varepsilon \rightarrow 0} \int_{-R+\varepsilon}^{R-\varepsilon} \sqrt{1+y^{\prime 2}} d x=\lim _{\varepsilon \rightarrow 0} \int_{-R+\varepsilon}^{R-\varepsilon} \frac{R d x}{\sqrt{R^{2}-x^{2}}}=\left.\lim _{\varepsilon \rightarrow 0} R \arcsin \frac{x}{R}\right|_{-R+\varepsilon} ^{R-\varepsilon}=\pi
$$