91Ó°ÊÓ

Write an integral that represents the area of the surface generated by revolving the curve about the \(x\) -axis. Use a graphing utility to approximate the integral. $$ x=4 t, \quad y=t+1 \quad 0 \leq t \leq 2 $$

A curve called the folium of Descartes can be represented by the parametric equations \(x=\frac{3 t}{1+t^{3}} \quad\) and \(y=\frac{3 t^{2}}{1+t^{3}}\) (a) Convert the parametric equations to polar form. (b) Sketch the graph of the polar equation from part (a). (c) Use a graphing utility to approximate the area enclosed by the loop of the curve.

Use a graphing utility to graph the polar equation and find all points of horizontal tangency. $$ r=2 \cos (3 \theta-2) $$

Write an integral that represents the area of the surface generated by revolving the curve about the \(x\) -axis. Use a graphing utility to approximate the integral. $$ x=\frac{1}{4} t^{2}, \quad y=t+2 \quad 0 \leq t \leq 4 $$

Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=3 \sin \theta $$

Write an integral that represents the area of the surface generated by revolving the curve about the \(x\) -axis. Use a graphing utility to approximate the integral. $$ x=\cos ^{2} \theta, \quad y=\cos \theta \quad 0 \leq \theta \leq \frac{\pi}{2} $$

Write an integral that represents the area of the surface generated by revolving the curve about the \(x\) -axis. Use a graphing utility to approximate the integral. $$ x=\theta+\sin \theta, \quad y=\theta+\cos \theta \quad 0 \leq \theta \leq \frac{\pi}{2} $$

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=3(1-\cos \theta) $$

Find the area of the surface generated by revolving the curve about each given axis. $$ \begin{array}{llll} x=t, y=2 t, & 0 \leq t \leq 4, & \text { (a) } x \text { -axis } & \text { (b) } y \text { -axis } \end{array} $$