5.4: Area and Arc Length in Polar Coordinates
- Page ID
- 163309
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Tangents to Polar Curves
Given \( x\left( \theta \right) = r \cos \theta \) and \( y\left( \theta \right) = r \sin \theta \), where \( r = f\left( \theta \right) \),\[ \dfrac{dy}{dx} = \dfrac{dy/d\theta}{dx/d\theta} = \dfrac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}. \nonumber \]
Determine the equation of the tangent line to \( r = 3 + 8 \sin \theta \) at \( \theta = \frac{\pi}{6} \). When are the tangent lines horizontal?
Areas "Under" Polar Curves
Suppose \(f\) is continuous and nonnegative on the interval \( \alpha \leq \theta \leq \beta \) with \(0< \beta − \alpha \leq 2 \pi \). The area of the region bounded by the graph of \(r=f( \theta )\) between the radial lines \( \theta = \alpha \) and \( \theta = \beta \) is\[ A =\dfrac{1}{2}\int ^ \beta _ \alpha [f( \theta )]^2 \, d \theta = \dfrac{1}{2}\int ^ \beta _ \alpha r^2 \, d \theta . \nonumber \]
Find the area of the region under \( r = \sqrt{\theta} \) from \( \theta = 0 \) to \( \theta = 2 \pi \).
Find the area of the region enclosed by the inner loop of the curve\[ r = 6 + 12 \sin \theta . \nonumber \]
\[ A = \int_{\alpha}^{\beta} \dfrac{1}{2} \left( r_{\text{outer}}^2 - r_{\text{inner}}^2 \right) \, d\theta \nonumber \]
Find the area of the region that lies inside the first curve and outside the second curve.\[ r = 15 \cos \theta, \quad r = 7 + \cos \theta \nonumber \]
Find the area of the region that lies simultaneously inside both curves.\[ r = \sin\left( 2\theta \right), \quad r = \cos\left( 2\theta \right) \nonumber \]
Arc Length of Polar Curves
Let \(f\) be a function whose derivative is continuous on an interval \( \alpha \leq \theta \leq \beta \). The length of the graph of \(r=f( \theta )\) from \( \theta = \alpha \) to \( \theta = \beta \) is\[ L =\int ^ \beta _ \alpha \sqrt{[f( \theta )]^2+[f^{\prime}( \theta )]^2}\,d \theta = \int ^ \beta _ \alpha \sqrt{r^2+\left(\dfrac{dr}{d \theta }\right)^2}\,d \theta . \nonumber \]
Find the arc length of \(r = 0\) for \( 0 \leq \theta \leq 2 \pi \).
Set up the integral for the length of the three-leaved rose \( r = \cos\left( 3 \theta \right) \), using the strategy of finding only the length from \( \theta = 0 \) to \( \theta = \frac{\pi}{6} \) first. Why is this integral difficult to compute? Approximate the value of this integral.