91Ó°ÊÓ

Find parametric equations and a parameter interval for the motion of a particle starting at the point \((2,0)\) and tracing the top half of the circle \(x^{2}+y^{2}=4\) four times.

Find the lengths of the curves in Exercises \(21-28 .\) The curve \(r=\cos ^{3}(\theta / 3), \quad 0 \leq \theta \leq \pi / 4\)

Exercises \(25-28\) give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation in Cartesian coordinates. $$\begin{array}{l}{\text { Eccentricity: } 3} \\ {\text { Foci: }( \pm 3,0)}\end{array}$$

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \cos \theta=2$$

Find the lengths of the curves. $$ x=t^{2} / 2, \quad y=(2 t+1)^{3 / 2} / 3, \quad 0 \leq t \leq 4 $$

Sketch the region defined by the inequality. \(0 \leq r \leq 2-2 \cos \theta\)

Find the lengths of the curves. $$ x=(2 t+3)^{3 / 2} / 3, \quad y=t+t^{2} / 2, \quad 0 \leq t \leq 3 $$

Sketch the region defined by the inequality. \(0 \leq r^{2} \leq \cos \theta\)

Find parametric equations and a parameter interval for the motion of a particle that moves along the graph of \(y=x^{2}\) in the following way: Beginning at \((0,0)\) it moves to \((3,9),\) and then travels back and forth from \((3,9)\) to \((-3,9)\) infinitely many times.

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \sin \theta=-1$$