Computation of Arc Length

Find the arc length of each of the following curves: $y=x^{\frac{3}{2}}$ ($0\leqslant x\leqslant 4$).

Find the arc length of each of the following curves: $y^{2}=2px$ ($0\leqslant x\leqslant x_{0}$).

Find the arc length of each of the following curves: $y=a\operatorname {ch}\dfrac {x}{a}$ from point $A(0,a)$ to point $B(b,h)$.

Find the arc length of each of the following curves: $y=e^{x}$ ($0\leqslant x\leqslant x_{0}$).

Find the arc length of each of the following curves: $x=\dfrac {1}{4}y^{2}-\dfrac {1}{2}\ln y$ ($1\leqslant y\leqslant e$).

Find the arc length of each of the following curves: $\displaystyle y=a\ln \dfrac {a^{2}}{a^{2}-x^{2}}$ ($0\leqslant x\leqslant b<a$).

Find the arc length of each of the following curves: $y=\ln \cos x$ ($0\leqslant x\leqslant a<\dfrac {\pi }{2}$).

Find the arc length of each of the following curves: $\displaystyle x=a\ln \dfrac {a+\sqrt{a^{2}-y^{2}}}{y}-\sqrt{a^{2}-y^{2}}$ ($0\leqslant b\leqslant y\leqslant a$).

Find the arc length of each of the following curves: $\displaystyle y^{2}=\dfrac {x^{3}}{2a-x}$ ($\displaystyle 0\leqslant x\leqslant \dfrac {5}{3}a$).

Find the arc length of each of the following curves: $\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ (astroid).

Find the arc length of each of the following curves: $\displaystyle x=\dfrac {c^{2}}{a}\cos^{3}t$, $\displaystyle y=\dfrac {c^{2}}{b}\sin^{3}t$, $c^{2}=a^{2}-b^{2}$ (evolute of an ellipse).

Find the arc length of each of the following curves: $x=a\cos^{4}t$, $y=a\sin^{4}t$.

Find the arc length of each of the following curves: $x=a(t-\sin t)$, $y=a(1-\cos t)$ ($0\leqslant t\leqslant 2\pi$).

Find the arc length of each of the following curves: $x=a(\cos t+t\sin t)$, $y=a(\sin t-t\cos t)$ ($0\leqslant t\leqslant 2\pi$) (involute of a circle).

Find the arc length of the following curve: $x=a(\operatorname {sh}t-t)$, $y=a(\operatorname {ch}t-1)$ ($0\leqslant t\leqslant T$).

Find the arc length of the following curve: $r=a\varphi$ (Archimedean spiral) ($0\leqslant \varphi \leqslant 2\pi$).

Find the arc length of the following curve: $r=ae^{m\varphi }$ ($m>0$) for $0<r<a$.

Find the arc length of the following curve: $r=a(1+\cos \varphi )$.

Find the arc length of the following curve: $\displaystyle r=\dfrac {p}{1+\cos \varphi }\quad \left(\left|\varphi \right|\leqslant \dfrac {\pi }{2}\right)$.

Find the arc length of the following curve: $r=a\sin^{3}\dfrac {\varphi }{3}$.

Find the arc length of the following curve: $r=a\operatorname {th}\dfrac {\varphi }{2}$ ($0\leqslant \varphi \leqslant 2\pi$).

Find the arc length of the following curve: $\displaystyle \varphi =\dfrac {1}{2}\left(r+\dfrac {1}{r}\right)\quad (1\leqslant r\leqslant 3)$.

Prove that the arc length of the ellipse

equals the length of one wave of the sine curve $y=c\sin \dfrac {x}{b}$, where $c=\sqrt{a^{2}-b^{2}}$.

The parabola $4ay=x^{2}$ rolls along the $Ox$ axis. Prove that the focus of the parabola traces out a catenary.

Find the ratio of the area enclosed by the loop

to the area of the circle whose circumference equals the perimeter of this loop's boundary curve.