{Use of Tech} Implicit function graph Explain and carry out a method for graphing the curve x = 1 + cos² y − sin² y using parametric equations and a graphing utility.

Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π

53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.

r = 1 - cos θ

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

An ellipse with vertices (±5, 0), passing through the point (4, 3/5)

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

A circle centered at the origin with radius 4, generated counterclockwise

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x

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

x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π