57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.


r = 3/(2 + cos θ)

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves

r = 1 + sin θ and r = 1 + cos θ

45–60. Areas of regions Find the area of the following regions.

The region inside one leaf of the rose r = cos 5θ

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola symmetric about the y-axis that passes through the point (2, -6)

45–60. Areas of regions Find the area of the following regions.

The region inside the limaçon r = 4 - 2 cos θ

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=t,y= √(4−t²) a