Year: 1939
Topic: Geometry Calculus

Problem:
A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3a$. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle.

Solution:
Take rectangular coordinates with the origin at the center of the large circle so that the generating point $P$ is in contact with the large circle at $A = (3a, 0)$. It can be seen from the diagram that when the small circle has rolled until the line of centers makes an angle $\theta$ with $OA$, the coordinates of $P$ are \[ x = 2a \cos \theta + a \cos 2\theta \] \[ y = 2a \sin \theta - a \sin 2\theta. \] These are, then, parametric equations for the path of $P$.
The area is given by \[ A = \frac{1}{2} \oint (x \ dy - y \ dx). \] \[ = \frac{a^2}{2} \int_0^{2\pi} \{ [2 \cos \theta + \cos 2\theta][2 \cos \theta - 2 \cos 2\theta] - [2 \sin \theta - \sin 2\theta][2 \sin \theta - 2 \sin 2\theta] \} d\theta \] \[ = \frac{a^2}{2} \int_0^{2\pi} (2 - 2 \cos 3\theta) d\theta = \boxed{2\pi a^2}. \]