Consider a solid formed by rotated a curve defined by the parametric equations \(x=x(t)\) and \(y=y(t)\) for \(a \le t \le b\) around the x-axis.
Suppose we want to find the surface area of this solid between \(t=a\) and \(t=b\). Let \(x(a) = \alpha\) and \(x(b) = \beta\). Finding the surface area between \(t=a\) and \(t=b\) is the same as finding the surface area between \(x = \alpha\) and \(x = \beta\):
\[S = 2 \pi \int_\alpha^\beta y \sqrt{1+ (\frac{dy}{dx})^2} \ dx\]
The integral with respect to \(t\) would be:
\[\begin{align} S &= 2 \pi \int_a^b y(t) \sqrt{1+ \left(\frac{dy/dt}{dx/dt} \right)^2}\ \frac{dx}{dt} \ dt \\ &= 2 \pi \int_a^b y(t) \sqrt{ \left( \frac{dx}{dt} \right)^2+ \left( \frac{dy}{dt} \right)^2 } \ dt \end{align}\]