OBJECTIVES

When you finish this section, you should be able to:

1. Graph a polar equation; find parametric equations (p. 671)
2. Find the arc length of a curve represented by a polar equation (p. 675)

Graphing a Polar Equation (Cardioid); Finding Parametric Equations

(a) Graph the polar equation \(r=1-\sin \theta,\;0\leq \theta\leq 2\pi.\)

(b) Find parametric equations for \(r=1-\sin \theta.\)

Solution (a) The polar equation \(r=1-\sin \theta\) contains \(\sin \theta\), which has the period \(2\pi .\) We construct Table 2 using common values of \(\theta\) that range from \(0\) to \(2\pi ,\) plot the points \(( r,\theta )\), and trace out the graph, beginning at the point \((1,0)\) and ending at the point \(( 1, \ 2\pi )\), as shown in Figure 43(a). Figure 43(b) shows the graph using technology.

Figure 43 The cardioid \(r=1-\sin \theta\).

(b) We obtain parametric equations for \(r=1-\sin \theta\) by using the conversion formulas \(x=r \ \cos \theta\) and \(y=r \ \sin\theta\): \[ x=r \ \cos \theta =( 1-\sin \theta ) \ \cos \theta \qquad y=r \ \sin \theta =( 1-\sin \theta ) \ \sin \theta \]

Here, \(\theta\) is the parameter, and if \(0\leq \theta \leq 2\pi ,\) then the graph is traced out exactly once in the counterclockwise direction.

NOW WORK

Problem 5.

NOTE

Limaçon (pronounced “leema sown”) is a French word for “snail.”

Graphing a Polar Equation (Limaçon Without an Inner Loop); Finding Parametric Equations

(a) Graph the polar equation \(r=3+2 \ \cos \theta,\;0\leq \theta\leq 2\pi.\)

(b) Find parametric equations for \(r=3+2 \ \cos \theta\).

Solution (a) The polar equation \(r=3+2 \ \cos \theta \) contains \(\cos \theta\), which has the period \(2\pi .\) We construct Table 3 using common values of \(\theta \) that range from \(0\) to \(2\pi ,\) plot the points \((r,\theta ) \), and trace out the graph, beginning at the point \((5,0) \) and ending at the point \(( 5, \ 2\pi ) \), as shown in Figure 44(a). Figure 44(b) shows the graph using technology.

Figure 44 The limaçon \(r=3+2 \ \cos \theta\).

(b) We obtain parametric equations for \(r=3+2 \ \cos \theta \) by using the conversion formulas \(x=r \ \cos \theta\) and \(y=r \ \sin \theta\): \[ x=r \ \cos \theta = ( 3+2 \ \cos \theta ) \ \cos \theta \qquad y=r \ \sin \theta =( 3+2 \ \cos \theta ) \ \sin \theta \]

Here, \(\theta\) is the parameter, and if \(0\leq \theta \leq 2\pi ,\) then the graph is traced out exactly once in the counterclockwise direction.

NOW WORK

Problem 7.

Graphing a Polar Equation (Limaçon with an Inner Loop); Finding Parametric Equations

(a) Graph the polar equation \(r=1+2 \ \cos \theta,\;0\leq \theta\leq 2\pi.\)

(b) Find parametric equations for \(r=1+2 \ \cos \theta \).

Solution (a) The polar equation \(r=1+2 \ \cos \theta\) contains \(\cos \theta\), which has the period \(2\pi .\) We construct Table 4 using common values of \(\theta \) that range from \(0\) to \(2\pi ,\) plot the points \(( r,\theta )\), and trace out the graph, beginning at the point \(\left(3,0\right)\) and ending at the point \(( 3, \ 2\pi )\), as shown in Figure 45(a). Figure 45(b) shows the graph using technology.

Figure 45 The limaçon \(r=1+2 \ \cos \theta\).

(b) We obtain parametric equations for \(r=1+2 \ \cos \theta\) by using the conversion formulas \(x=r \ \cos \theta\) and \(y=r \ \sin \theta\): \[ x=r \ \cos \theta =( 1+2 \ \cos \theta ) \ \cos \theta \qquad y=r \ \sin \theta =( 1+2 \ \cos \theta ) \ \sin \theta \]

Here, \(\theta\) is the parameter, and if \(0\leq \theta \leq 2\pi ,\) then the graph is traced out exactly once in the counterclockwise direction.

NOW WORK

Problem 9.

Graphing a Polar Equation (Rose); Finding Parametric Equations

(a) Graph the polar equation \(r=2 \ \cos ( 2\theta ) ,\) \(0\leq \theta \leq 2\pi\).

(b) Find parametric equations for \(r=2 \ \cos ( 2\theta )\).

Solution (a) The polar equation \(r=2 \ \cos ( 2\theta )\) contains \(\cos ( 2\theta )\), which has the period \(\pi\). So, we construct Table 5 using common values of \(\theta\) that range from \(0\) to \(2\pi\), noting that the values for \(\pi \leq \theta \leq 2\pi\) repeat the values for \(0\leq \theta \leq \pi .\) Then we plot the points \(( r,\theta)\) and trace out the graph. Figure 46(a) on page 674 shows the graph from the point \(( 2,0)\) to the point \(( 2, \ \pi).\) Figure 46(b) completes the graph from the point \(( 2, \ \pi)\) to the point \(( 2, \ 2\pi )\).

Figure 46 A rose with four petals.

(b) Parametric equations for \(r=2 \ \cos ( 2\theta )\): \begin{equation*} x=r \ \cos \theta =2 \ \cos ( 2\theta ) \ \cos \theta \qquad y=r \ \sin \theta =2 \ \cos ( 2\theta ) \ \sin \theta \end{equation*}

where \(\theta\) is the parameter, and if \(0\leq \theta \leq 2\pi ,\) then the graph is traced out exactly once in the counterclockwise direction.

NOW WORK

Problem 11.

Graphing a Polar Equation (Spiral); Finding Parametric Equations

(a) Graph the equation \(r=e^{\theta /5}\).

(b) Find parametric equations for \(r=e^{\theta /5}\).

Solution The polar equation \(r=e^{\theta /5}\) lacks the symmetry you may have observed in the previous examples. Since there is no number \(\theta\) for which \(r=0\), the graph does not contain the pole. Also observe that:

• \(r\) is positive for all \(\theta\).
• \(r\) increases as \(\theta\) increases.
• \(r\rightarrow 0\) as \(\theta \rightarrow -\infty\).
• \(r\rightarrow \infty\) as \(\theta \rightarrow \infty \).

We use a calculator to obtain Table 6. Figure 47(a) shows part of the graph \(r=e^{\theta /5}.\) Figure 47(b) shows the graph for \(\theta =-\dfrac{3\pi}{2}\) to \(\theta =2\pi\) using technology.

Figure 47 The spiral \(r=e^{\theta /5}\).

(b) We obtain parametric equations for \(r=e^{\theta /5}\) by using the conversion formulas \(x=r \ \cos \theta\) and \(y=r \ \sin \theta\): \[ x=r \ \cos \theta =e^{\theta /5} \ \cos \theta \qquad y=r \ \sin \theta =e^{\theta /5}\sin \theta \]

where \(\theta\) is the parameter and \(\theta\) is any real number.

THEOREM Arc Length of the Graph of a Polar Equation

If a curve \(C\) is represented by the polar equation \(r=f(\theta), \alpha \leq \theta \leq \beta\), and if \(f' (\theta )=\dfrac{dr}{d\theta }\) is continuous on an interval containing \(\alpha\) and \(\beta\), then the arc length \(s\) of \(C\) from \(\theta =\alpha\) to \(\theta =\beta\) is \[\bbox[5px, border:1px solid black, #F9F7ED]{\bbox[#FAF8ED,5pt]{ s=\int_{\alpha }^{\beta }\sqrt{\,r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}}\,d\theta}} \]

Finding the Arc Length of a Logarithmic Spiral

Find the arc length \(s\) of the logarithmic spiral represented by \(r=f(\theta)=e^{3\theta }\) from \(\theta = 0\) to \(\theta =2.\)

Solution We use the arc length formula \(s=\int_{\alpha }^{\beta } \sqrt{\,r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}}\,d\theta\) with \(r=e^{3\theta }\). Then \(\dfrac{dr}{d\theta }=3e^{3\theta }\) and \begin{eqnarray*} s&=&\int_{0}^{2}\sqrt{(e^{3\theta })^{2}+(3e^{3\theta })^{2}}\,d\theta =\int_{0}^{2}\sqrt{10e^{6\theta }}d\theta =\sqrt{10}\int_{0}^{2}e^{3\theta }\,d\theta\\ &=& \sqrt{10}\left[\dfrac{e^{3\theta}}{3}\right]^2_0 =\frac{\sqrt{10}}{3}(e^{6}-1) \end{eqnarray*}

NOW WORK

Problem 19.