91Ó°ÊÓ

Sketch the curve in polar coordinates. $$ r=-2 \cos 2 \theta $$

Find the area of the region described. $$ \begin{array}{l}{\text { The region inside the cardioid } r=2+2 \cos \theta \text { and to the }} \\ {\text { right of the line } r \cos \theta=\frac{3}{2} \text { . }}\end{array} $$

Sketch the curve in polar coordinates. $$ r=3 \sin 2 \theta $$

Find \(d y / d x\) and \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. $$ x=\sqrt{t}, y=2 t+4 ; t=1 $$

Sketch the curve in polar coordinates. $$ r=9 \sin 4 \theta $$

Find the area of the region described. $$ \begin{array}{l}{\text { The region inside the circle } r=2 \text { and to the right of the line }} \\ {r=\sqrt{2} \sec \theta}\end{array} $$

Find \(d y / d x\) and \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. $$ x=\frac{1}{2} t^{2}+1, \quad y=\frac{1}{3} t^{3}-t ; t=2 $$

Sketch the curve in polar coordinates. $$ r=2 \cos 3 \theta $$

Find the area of the region described. $$ \begin{array}{l}{\text { The region inside the rose } r=2 a \cos 2 \theta \text { and outside the }} \\ {\text { circle } r=a \sqrt{2} \text { . }}\end{array} $$

A nuclear cooling tower is to have a height of \(h\) feet and the shape of the solid that is generated by revolving the region \(R\) enclosed by the right branch of the hyperbola \(1521 x^{2}-225 y^{2}=342,225\) and the lines \(x=0\), \(y=-h / 2,\) and \(y=h / 2\) about the \(y\) -axis. (a) Find the volume of the tower. (b) Find the lateral surface area of the tower.