Calculus 2: Area of a surface of revolution

Find the surface area of the solid formed by revolving the curve $y=\sqrt{4x-x^2}$ about the x-axis, where x ranges from $0.5$ to $1.5$.

Find the surface area of the solid formed by revolving the curve $x = 3\sqrt{4 - y}$ about the y-axis, where $y$ ranges from 0 to $\frac{15}{4}$.

Calculate the surface area of a solid of revolution when rotating a given curve segment about the x-axis using the formula $S = \int_{a}^{b} 2\pi y \, \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$ or $S = \int_{c}^{d} 2\pi y \, \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy$, depending on whether the curve is described as $y(x)$ or $x(y)$.

Given a function that forms a squiggly line, rotate it around the x-axis to form a three-dimensional shape. Find the surface area of this shape.

Find the surface area of the curve $x^3$ rotated around the x-axis from $x = 0$ to $x = 1$.

Find the surface area of revolution of the function f(x) = x^3 around the x-axis from x = 0 to x = 1.

Compute the surface area of a region of revolution, specifically Gabriel's horn for $\frac{1}{x}$.