Chapter 13

Find the area of the region enclosed by the graph of the given equation.\(r=3 \cos \theta\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{c}2 r=3 \\ r=3 \sin \theta\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(r \cos \theta=4\)

Find the area of the region enclosed by the graph of the given equation.\(r=2-\sin \theta\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}2 r=3 \\ r=1+\cos \theta\end{array}\right.\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=2 \cos \theta \\ r=2 \sin \theta\end{array}\right.\)

Plot the point having the given set of polar coordinates; then find another set of polar coordinates for the same point for which (a) \(r<0\) and \(0 \leq \theta<2 \pi ;\) (b) \(r>0\) and \(-2 \pi<\theta \leq 0 ;\) (c) \(r<0\) and \(-2 \pi<\theta \leq 0\).\(\left(2, \frac{1}{2} \pi\right)\)

Find the area of the region enclosed by the graph of the given equation.\(r=4 \sin ^{2} \frac{1}{2} \theta\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=2 \cos 2 \theta \\ r=2 \sin \theta\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(r=2 \sin \theta\)

Plot the point having the given set of polar coordinates; then find another set of polar coordinates for the same point for which (a) \(r<0\) and \(0 \leq \theta<2 \pi ;\) (b) \(r>0\) and \(-2 \pi<\theta \leq 0 ;\) (c) \(r<0\) and \(-2 \pi<\theta \leq 0\).\(\left(3, \frac{3}{2} \pi\right)\)

Find the area of the region enclosed by the graph of the given equation.\(r^{2}=4 \sin 2 \theta\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=4 \theta \\ r=\frac{1}{2} \pi\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(\theta=5\)

Find the area of the region enclosed by the graph of the given equation.\(r=4 \sin ^{2} \theta \cos \theta\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}\theta=\frac{1}{6} \pi \\\ r=2\end{array}\right.\)

Find the area of the region enclosed by one loop of the graph of the given equation.\(r=3 \cos 2 \theta\)

Draw a sketch of the graph of the given equation.\(r=5\)

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\(\left(3,-\frac{2}{3} \pi\right)\)

Find the area of the region enclosed by one loop of the graph of the given equation.\(r=a \sin 3 \theta\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=1-\sin \theta \\ r=\cos 2 \theta\end{array}\right.\)

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\(\left(\sqrt{2},-\frac{1}{4} \pi\right)\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=\sin 2 \theta \\ r=\cos 2 \theta\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(r \sin \theta=-4\)

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\(\left(-4, \frac{5}{6} \pi\right)\)

Draw a sketch of the graph of the given equation.\(r \cos \theta=-5\)

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\(\left(-2, \frac{4}{3} \pi\right)\)

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{l}r=2 \\ r=3-2 \cos \theta\end{array}\right.\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=\tan \theta \\ r=4 \sin \theta\end{array}\right.\)

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{l}r=4 \sin \theta \\ r=4 \cos \theta\end{array}\right.\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=4 \tan \theta \sin \theta \\ r=4 \cos \theta\end{array}\right.\)

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\((-3,-\pi)\)

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{l}r=3 \sin 2 \theta \\ r=3 \cos 2 \theta\end{array}\right.\)

Find a measurement of the angle between the tangent lines of the given pair of curves at all points of intersection.\(\left\\{\begin{array}{l}r=1-\sin \theta \\\ r=1+\sin \theta\end{array}\right.\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=4(1+\sin \theta) \\ r(1-\sin \theta)=3\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(r=e^{\theta}\) (logarithmic spiral)

Find the rectangular cartesian coordinates of each of the following points whose polar coordinates are given: (a) \((3, \pi)\); (b) \(\left(\sqrt{2},-\frac{3}{4} \pi\right) ;\) (c) \(\left(-4, \frac{2}{3} \pi\right) ;\) (d) \(\left(-2,-\frac{1}{2} \pi\right) ;\) (e) \(\left(-2, \frac{7}{4} \pi\right) ;\) (f) \(\left(-1,-\frac{\tau}{6} \pi\right)\).

Find the area of the intersection of the regions enclosed by the graphs of the two given equations.\(\left\\{\begin{array}{c}r^{2}=2 \cos 2 \theta \\\ r=1\end{array}\right.\)

Find a measurement of the angle between the tangent lines of the given pair of curves at all points of intersection.\(\left\\{\begin{array}{l}r=3 \cos \theta \\\ r=1+\cos \theta\end{array}\right.\)

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{r}r^{2} \sin 2 \theta=8 \\ r \cos \theta=2\end{array}\right.\)

Find a set of polar coordinates for each of the following points whose rectangular cartesian coordinates are given. Take \(r>0\) and \(0 \leq \theta<2 \pi\). (a) \((1,-1) ;\) (b) \((-\sqrt{3}, 1) ;(\mathrm{c})(2,2) ;\) (d) \((-5,0) ;\) (e) \((0,-2) ;\) (f) \((-2,-2 \sqrt{3})\).

Find a measurement of the angle between the tangent lines of the given pair of curves at all points of intersection.\(\left\\{\begin{array}{l}r=\cos \theta \\\ r=\sin 2 \theta\end{array}\right.\)

The graph of the given equation intersects itself. Find the points at which this occurs.\(r=\sin \frac{3}{2} \theta\)

Draw a sketch of the graph of the given equation.\(r=1 / \theta\) (reciprocal spiral)

Find a polar equation of the graph having the given cartesian equation.\(x^{2}+y^{2}=a^{2}\)

Find the area of the region which is inside the graph of the first equation and outside the graph of the second equation.\(\left\\{\begin{array}{l}r=2 a \sin \theta \\ r=a\end{array}\right.\)

The graph of the given equation intersects itself. Find the points at which this occurs.\(r=1+2 \cos 2 \theta\)

Draw a sketch of the graph of the given equation.\(r=2 \theta\) (spiral of Archimedes)

Find the area of the region which is inside the graph of the first equation and outside the graph of the second equation.\(\left\\{\begin{array}{l}r=2 \sin \theta \\ r=\sin \theta+\cos \theta\end{array}\right.\)

Draw a sketch of the graph of the given equation.\(r=2 \sin 3 \theta\) (three- leafed rose)