Question

Find a polar equation for the curve represented by the given Cartesian equation. $$ y=-2 x^{2} $$

Name: Problem 24, Chapter 10: Calculus: Early Transcendentals, Metric Edition
Uploaded: 2022-09-14T15:45:55-08:00
Duration: 1 min 57 s
Channel: Lauren Shelton
Description: Find a polar equation for the curve represented by the given Cartesian equation. y=-2 x^2

   Find a polar equation for the curve represented by the given Cartesian equation.
$$
y=-2 x^{2}
$$

Calculus: Early Transcendentals, Metric Edition

James Stewart,… 9th Edition

Chapter 10, Problem 24

Instant Answer

Solved by Expert Lauren Shelton

09/14/2022

Step 1

We know that in polar coordinates, $x = r\cos(\theta)$ and $y = r\sin(\theta)$. So, we can substitute these into the Cartesian equation to get $r\sin(\theta) = -2(r\cos(\theta))^{2}$. Show more…

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Find a polar equation for the curve represented by the given Cartesian equation. $$ y=-2 x^{2} $$

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Key Concepts

Polar Coordinates

Polar coordinates are a system for representing points in the plane using a distance from a fixed origin (r) and an angle (?) measured from the positive x-axis. This system is especially useful for dealing with curves and problems that exhibit rotational symmetry or where radial distance and angular direction are more natural descriptors than horizontal and vertical positions.

Conversion from Cartesian to Polar Equations

This process involves substituting the expressions for x and y in Cartesian coordinates (x = r*cos(?) and y = r*sin(?)) into the given Cartesian equation. After substitution, algebraic manipulations are carried out to express the relationship in terms of r and ?. This method not only simplifies certain curves but also provides new insights into the characteristics of the curve based on the interplay between the radius and the angle.

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