   Chapter 10.5, Problem 83E

Chapter
Section
Textbook Problem

Folium of Descartes A curve called the folium of Descartes can be represented by the parametric equations x = 3 t 1 + t 3 and y = 3 t 2 1 + t 3 (a) Convert the parametric equations to polar form.(b) Sketch the graph of the polar equation from part (a).(c) Use a graphing utility to approximate the area enclosed by the loop of the curve.

(a)

To determine

To Calculate: Toconvert theparametric equationinto polar form.

Explanation

Given:

The parametric equations x=3t1+t3 and y=3t21+t3.

Formula Used:

x=f(t),y=g(t)

Calculation:

Simplify the equation:

x3+y3=(3t1+t3)3+(3t21+t3)3=27t3(1+t3)3+27t6(1+t3)3=27t3(1+t3)(1+t3)3=27t3(1+t3)2=3.(3t1+t3)

(b)

To determine

To Graph: TheGraph for r=3cosθsinθ(cos3θ+sin3θ).

(c)

To determine

To Calculate: The area enclosed by curve by using graph utility in part (b).

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