91Ó°ÊÓ

In Exercises \(19-24,\) match the parametric equations with the parametric curves labeled A through F. $$ x=\cos t, \quad y=\sin 3 t $$

In Exercises \(17-24,\) find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$ 64 x^{2}-36 y^{2}=2304 $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$-\pi / 4 \leq \theta \leq \pi / 4, \quad-1 \leq r \leq 1$$

Find the area under \(y=x^{3}\) over \([0,1]\) using the following parametrizations. $$ \quad a. x=t^{2}, \quad y=t^{6} \quad \text { b. } x=t^{3}, \quad y=t^{9} $$

Give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. \(169 x^{2}+25 y^{2}=4225\)

Find the lengths of the curves in Exercises \(21-28\) . The curve \(r=a \sin ^{2}(\theta / 2), \quad 0 \leq \theta \leq \pi, \quad a>0\)

Find the lengths of the curves in Exercises \(21-28\) . The parabolic segment \(r=6 /(1+\cos \theta), \quad 0 \leq \theta \leq \pi / 2\)

Give information about the foci and vertices of ellipses centered at the origin of the \(x y\)-plane. In each case, find the ellipse's standard-form equation from the given information. Foci: \(( \pm \sqrt{2}, 0)\) Vertices: \(( \pm 2,0)\)

Find the lengths of the curves in Exercises \(25-30 .\) $$ x=\cos t, \quad y=t+\sin t, \quad 0 \leq t \leq \pi $$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$-\pi / 2 \leq \theta \leq \pi / 2, \quad 1 \leq r \leq 2$$