91Ó°ÊÓ

\(37-40\) Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $$ x=t+e^{-t}, \quad y=t-e^{-t}, \quad 0 \leq t \leqslant 2 $$

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r=2 \cos 4 \theta\)

Compare the curves represented by the parametric equations. How do they differ? $$ \begin{array}{ll}{\text { (a) } x=t,} & {y=t^{-2}} \\ {\text { (c) } x=e^{t},} & {y=e^{-2 t}}\end{array} \quad \text { (b) } x=\cos t, \quad y=\sec ^{2} t $$

Find all points of intersection of the given curves. $$ r=1+\cos \theta, \quad r=1-\sin \theta $$

Find an equation for the conic that satisfies the given conditions. $$ \text { Ellipse, foci }(0, \pm \sqrt{2}), \text { vertices }(0, \pm 2) $$

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r=2 \sin 6 \theta\)

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $$ x=t^{2}-t, \quad y=t^{4}, \quad 1 \leqslant t \leqslant 4 $$

Find all points of intersection of the given curves. $$ r=2 \sin 2 \theta, \quad r=1 $$

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r\)r=1+3 \cos \theta$

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $$ x=t-2 \sin t, \quad y=1-2 \cos t, \quad 0 \leqslant t \leqslant 4 \pi $$