Calculus with Parametric Curves: Surface Areas

A parametric curve is a description of a path in the plane where both the x and y coordinates are functions of a single parameter, typically denoted t. When such a curve is revolved about an external axis lying in the same plane, it generates a surface of revolution in three dimensions. The surface area of this rotated shape depends fundamentally on two aspects: the geometry of the original curve and how far it lies from the chosen axis of rotation.

A torus is a classical surface of revolution that arises when a circle is rotated about an axis in its plane that does not intersect the circle. Consider a circle of radius r whose center lies at a distance R from a fixed axis of rotation. As the circle rotates, every point on it traces a circular path in space. The resulting solid has a ring-like geometry with a central void; the radius r controls the thickness of the torus’s tube, while R determines the size of the toroidal “hole.”

Mathematically, if the circle lies in the xy-plane and is centered at (R, 0), it can be parameterized as: x(t) = R + rcos t and y(t) = rsin t, with t varying from 0 to 2𝜋. If this curve is revolved about the y-axis, the torus is generated.

To derive the surface area of the torus, one considers an infinitesimal strip on the original curve. This strip has a width given by the differential of arc length,

\begin{equation*}dS=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\end{equation*}

and it sweeps out a circular “ring” when rotated. The radius of this ring from the axis of rotation is x(t), so the length of the circular path traced by the strip is 2𝜋x(t). The differential surface area element dS is then the product of this path length and the strip width. Integrating over the parameter t from 0 to 2𝜋 yields the total surface area:

\begin{equation*}S= \int_{0}^{2\pi} 2\pi x(t)\,\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\end{equation*}

For a torus, this evaluates to: S = 4𝜋²Rr, highlighting how both the tube radius r and the distance R from the axis influence the final surface area.