Exploring Parametric Curves: Finding the Area of an Inner Loop
Can you calculate the area enclosed by one of the inner loops formed by the curve and ?
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Table of Contents
The Problem
From VOL. 55, NO. 22, MARCH 2024 edition of THE COLLEGE MATHEMATICS JOURNAL:
Find the area enclosed by one of the four inner loops.
— Greg Dresden, Washington and Lee University, Lexington, VA
Breaking Down the Problem
This problem revolves around a parametric curve, with equations for
Parametric Equations for the Curve
The parametric equations for the curve are:

Notice that the point closest to the origin on the inner loop in the first quadrant is given by
From the symmetry of the curve,
Since

Solving for and
Now, let’s examine the points of self-intersection we can apply the multiple-angle formulas to rewrite the parametric equations in terms of trigonometric identities:
Thus, the parametric equations become:
Dividing both sides by
To simplify further, let
This expands to:
which factors as:
Given that
Since
Using the quadratic formula, we find
and
Since the discriminant
and the solutions for
For future reference, it’s important to notice that:
and
Finding the Area
To compute the area enclosed by one of the inner loops, we use the parametric area formula:
Substituting the parametric expressions for
Using the values of
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This problem elegantly ties together parametric equations, trigonometric identities, and calculus. Through careful manipulation and integration, we arrived at a precise answer for the area (as well as a numerical approximation).