Chapter 10, Problem 62E

A curve called the folium of Descartes is defined by the parametric equations

$x=\frac{3t}{1+t^3}y=\frac{3t^2}{1+t^3}$

(a) Show that if (a, b) lies on the curve, then so does (b, a); that is, the curve is symmetric with respect to the line y = x. Where does the curve intersect this line?

(b) Find the points on the curve where the tangent lines are horizontal or vertical.

(c) Show that the line y = −x − 1 is a slant asymptote.

(d) Sketch the curve.

(e) Show that a Cartesian equation of this curve is x³ + y³ = 3xy.

(f) Show that the polar equation can be written in the form

$r=\frac{3\sec \theta \tan \theta }{1+{\tan }^3\theta }$

(g) Find the area enclosed by the loop of this curve.

(h) Show that the area of the loop is the same as the area that lies between the asymptote and the infinite branches of the curve. (Use a computer algebra system to evaluate the integral.)