85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.

15–22. Sets in polar coordinates Sketch the following sets of points.

r = 3

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.

x = 4 cos t, y = 4 sin t; slope = 1/2

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

x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = 8 + 2t, y = 1; −∞ < t < ∞

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 1 - sin θ; (1/2, π/6)

Plot the points with polar coordinates (2, π/6) and (−3, −π/2). Give two alternative sets of coordinate pairs for both points.