13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


4x² - y² = 16

81–88. Arc length Find the arc length of the following curves on the given interval.

x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π

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

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the circle r = 8 sin θ

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π. 

r = 3/(1 - cos θ)

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

The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ

63–74. Arc length of polar curves Find the length of the following polar curves.

{Use of Tech} The complete limaçon r=4−2cosθ