Q2Mathematics I

To calculate the Surface Area of Revolution (specifically when revolved about the horizontal x-axis), we utilize the principles of integral calculus. Consider a continuous Cartesian curve described by the explicit function $y=f(x)$ that is smoothly revolved completely around the x-axis, bounded strictly between the vertical limits $x=a$ and $x=b$.

The mathematical formula for computing this total generated surface area $S$ is:

In this strict mathematical formulation: - $y=f(x)$ actively represents the precise radius of the circular cross-section at any given horizontal point $x$. - The differential term $ds=\sqrt{1+(dy/dx{)}^2}dx$ rigorously represents the microscopic arc length of the actual curve itself. - The term $2\pi y$ represents the physical circumference of the circular strip generated during the revolution. - $a$ and $b$ strictly define the absolute boundary limits of integration along the x-axis.