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Essential Calculus: Early Transcendentals
Found in: Page 522
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Find a polar equation for the curve represented by the given Cartesian equation.

\({\rm{4}}{{\rm{y}}^{\rm{2}}}{\rm{ = x}}\)

The substitution equation

\({\rm{r = }}\frac{{{\rm{cos\theta }}}}{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\)

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Formulae that translate.

Formulas’ expressing a point’s Cartesian coordinates \(\left( {{\rm{x,y}}} \right)\)in terms of polar coordinates\(\left( {{\rm{r,\theta }}} \right){\rm{.}}\)

\(\begin{aligned}{l}{\rm{x = rcos\theta }}\\{\rm{y = rsin\theta }}\end{aligned}\)

Step 2:  Replace that in the equation.        

\(\begin{aligned}{c}{\rm{4}}{{\rm{y}}^{\rm{2}}}{\rm{ = x4 \times (rsin\theta }}{{\rm{)}}^{\rm{2}}}{\rm{ }}\\{\rm{ = rcos\theta 4}}{{\rm{r}}^{\rm{2}}}{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}\\{\rm{ = rcos\theta 4rsi}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}\\{\rm{ = cos\theta r }}\\{\rm{ = }}\frac{{{\rm{cos\theta }}}}{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\end{aligned}\)

In the equation, substitute that\({\rm{r = }}\frac{{{\rm{cos\theta }}}}{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\)

Most popular questions for Math Textbooks

Sketch the curve with the given polar equation by first sketching the graph of as a function of\({\rm{\theta }}\) in Cartesian coordinates.

\({\rm{r = \theta ,\theta > 0}}\)

Find the area of the region that is bounded by the given curve and lies in the specified sector.

\({\rm{r = }}{{\rm{e}}^{{\raise0.5ex\hbox{$\scriptstyle {{\rm{ - \theta }}}$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle {\rm{4}}$}}}}{\rm{,}}\frac{{\rm{\pi }}}{{\rm{2}}} \le {\rm{\theta }} \le {\rm{\pi }}\)

Find the exact length of the polar curve.

\({\rm{r}} = {{\rm{e}}^{{\rm{2\theta }}}}{\rm{,0}} \le {\rm{\theta }} \le {\rm{2\pi }}\)

Write an expression for the length of a parametric curve.

Find the area of the region enclosed by one loop of the curve.

\({\rm{r = 1 + 2sin\theta }}\)
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