91Ó°ÊÓ

Find the area of the region enclosed by the astroid \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta .\) (Astroids are explored in the Laboratory Project on page \(649 .\) )

Find an equation for the conic that satisfies the given conditions. $$ \text { Parabola, focus }(2,-1), \quad \text { vertex }(2,3) $$

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r=\theta^{2},-2 \pi \leqslant \theta \leqslant 2 \pi\)

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

Find an equation for the conic that satisfies the given conditions. $$ \begin{array}{l}{\text { Parabola, vertex }(3,-1), \text { horizontal axis, }} \\\ {\text { passing through }(-15,2)}\end{array} $$

Find an equation for the conic that satisfies the given conditions. $$ \begin{array}{l}{\text { Parabola, vertical axis, }} \\ {\text { passing through }(0,4),(1,3), \text { and }(-2,-6)}\end{array} $$

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. \(r=-\sin 5 \theta\)

\(37-42\) Find all points of intersection of the given curves. $$ r=\sin \theta, \quad r=1-\sin \theta $$

Find an equation for the conic that satisfies the given conditions. $$ \text { Ellipse, foci }(\pm 2,0), \quad \text { vertices }(\pm 5,0) $$

Compare the curves represented by the parametric equations. How do they differ? $$ \begin{array}{ll}{\text { (a) } x=t^{3},} & {y=t^{2}} \\ {\text { (c) } x=e^{-3 t},} & {y=e^{-2 t}}\end{array} \quad \text { (b) } x=t^{6}, \quad y=t^{4} $$