# Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.2 Arc Length and Speed - Exercises - Page 611: 12

$$\pi \sqrt{4\pi ^2+1}+\frac{1}{2}\ln \left(2\pi +\sqrt{4\pi ^2+1}\right)$$

#### Work Step by Step

\begin{aligned} s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\ &=\int_{0}^{2 \pi} \sqrt{ (\cos \left(t\right)-t\sin \left(t\right))^2+(\sin \left(t\right)+t\cos \left(t\right))^2} d t \\ &= \int_{0}^{2 \pi} \sqrt{ t^2\cos ^2\left(t\right)+t^2\sin ^2\left(t\right)+1} d t\\ &= \int_{0}^{2 \pi} \sqrt{ t^2 +1} d t\\ &= \frac{1}{2} t \sqrt{1+t^{2}}+\frac{1}{2} \ln (t+\sqrt{1+t^{2}})\bigg|_{0}^{2\pi}\\ &= \pi \sqrt{4\pi ^2+1}+\frac{1}{2}\ln \left(2\pi +\sqrt{4\pi ^2+1}\right) \end{aligned}

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