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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

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

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