91Ó°ÊÓ

In this section we graphed many polar equations, and

most of our examples followed similar procedures. Describe

a general procedure for sketching the graph of a

polar equation.

Use polar coordinates to graph each of the following functions.

$r=2-2\sin \mathrm{θ}$

sketch the parametric curve by plotting points.

$x=3t+1,y=1,t∈[-2,5]$

Let $\mathrm{Î\pm },\mathrm{β}$, and $\mathrm{γ}$ be nonzero constants. Show that the graph of $r=\frac{\mathrm{Î\pm }}{\mathrm{β}+\mathrm{γ}\sin \mathrm{θ}}$ is a conic section with eccentricity $\left|\frac{\mathrm{γ}}{\mathrm{β}}\right|$ and directrix $y=\frac{\mathrm{Î\pm }}{\mathrm{γ}}$.

Find a definite integral expression that represents the area of the given region in the polar plane and then find the exact value of expression

The region bounded enclosed by the spiral$r=\mathrm{θ}$and the x-axis on the interval$0≤\mathrm{θ}≤\mathrm{π}$

Graph the equation in the θr-plane. Label each arc of your curve with the quadrant in which the corresponding polar graph will occur.

$r=1-\cos \mathrm{θ}$

The polar coordinates for several sets of points are given. Find the rectangular coordinates for each of the points, and then plot and label the points in the same polar coordinate system.

$\left(3,\frac{\mathrm{Ï€}}{6}\right),\left(-3,\frac{\mathrm{Ï€}}{6}\right),\left(3,-\frac{\mathrm{Ï€}}{6}\right)$and$\left(-3,-\frac{\mathrm{Ï€}}{6}\right)$

In Exercises 17–25 find a definite integral expression that represents

the area of the given region in the polar plane, and

then find the exact value of the expression.

The region enclosed by the spiral $r=\mathrm{θ}$and the $x$-axis on

the interval $0≤\mathrm{θ}≤\mathrm{π}$

Complete the square to describe the conics in Exercises $18-21$.

$2x^2+4x+y^2-6y-3=0$

sketch the parametric curve by plotting points. $x=3+1,y=2t,t∈⌊0,s⌋$