MATHEMATICS 121, Problem Set F
1) Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates for that
point, one with r > 0 and one with r < 0.
a) 1, π2
b) (−1, π)
2) Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.
a) (4, 3π)
b) − 2, − 65 π
3) Find the polar coordinates, with r > 0 and 0 ≤ θ ≤ 2π, of the point whose Cartesian coordinates are
(−1, 1).
4) Sketch the region in the plane consisting of points whose polar coordinates obey 1 ≤ r < 3, − π4 ≤ θ ≤
π
4.
5) Find the distance between the points whose polar coordinates are (r1 , θ1 ) and (r2 , θ2 ).
6) Find a Cartesian equation for the curve with the given polar equation
a) r = 2 sin θ
b) r =
5
3−4 sin θ
7) Find a polar equation for the curve with Cartesian equation x2 = 4y
8) Sketch the curves with the given polar equations.
a) r = 2 sin θ + 2 cos θ
c) r = eθ
b) r = 1 + cos θ
9) Find the slope of the tangent line to r = cos θ + sin θ at θ =
π
4.
10) Find the points where the tangent line to r = cos θ + sin θ is horizontal or vertical.
11) Suppose that ab 6= 0. Show that r = a sin θ + b cos θ represents a circle and find its center and radius.
12) Match the polar equations with the graphs (labelled I-VI). Give reasons for your choices.
a) r = sin 2θ
b) r = sin 4θ
c) r = sec(3θ)
d) r = θ sin θ
e) r = 1 + 4 cos 5θ
f) r =
I
IV
II
V
√1
θ
III
VI
13) Find the area of the region formed by the curve parametrized by the equations x = 4a cos t, y =
√
a 2 sin t cos t for − π2 ≤ t ≤ π2 .
14) Find the area of the region bounded by r = eθ and with − π2 ≤ θ ≤
π
2.
15) Sketch r = sin 3θ and find the area that it encloses.
16) Find the area of the region that is inside r = 3 cos θ and is outside r = 2 − cos θ.
17) Find the area of the region that lies inside both r = sin 2θ and r = cos 2θ.
18) Find the area of the region that lies inside both r 2 = 2 sin 2θ and r = 1.
19) Find the lengths of
a) r = e−θ , 0 ≤ θ ≤ 3π
b) r = θ, 0 ≤ θ ≤ 2π
20) Consider the cardiod r = 1 − cos θ and the unit circle r = 1.
a) Graph both curves on the same graph.
b) Find the area which lies inside the cardioid, but outside the circle.
21) Sketch and find the area of the plane region that lies inside both the circle with polar equation r = sin θ
√
and the circle with polar equation r = 3 cos θ.
22) Find the area of the region lying inside both r = 1 and r = 2 sin 2θ.
Reminder: Quiz V is on Tuesday, April 1. It will cover up to the end of the class of Friday, March 28.
Reminder: The final exam is on Tuesday, April 15 at 8:30 a.m. in HENN 202.