11.5 SUMMARY

• Arc length of \(c(t) = (x(t),y(t))\) for \(a \le t \le b\): \[ s = \textrm{arc length} = \int_a^b \sqrt{x'(t)^2 + y'(t)^2}\, dt \]
• The arc length is the distance along the path \(c(t)\). The displacement is the distance from the starting point \(c(a)\) to the endpoint \(c(b)\).
• Arc length integral: \[{s(t) = \int_{t_0}^t \sqrt{x'(u)^2 + y'(u)^2}\,du}\]
• Speed at time \(t\): \[{ \frac{ds}{dt} = \sqrt{x'(t)^2 + y'(t)^2}}\]
• Surface area of the surface obtained by rotating \(c(t) = (x(t),y(t))\) about the \(x\)-axis for \(a\le t \le b\): \begin{equation*} S = 2\pi \int_a^b y(t)\sqrt{x'(t)^2+y'(t)^2}\,dt \end{equation*}