The surface formed by rotating the curve \( y = \frac{1}{x} \), \( x \geq 1 \), about the x-axis is known as Gabriel's horn. Show that the surface area is infinite (although the enclosed volume is finite).**Diagram Description:**The image shows a 3D representation of Gabriel's horn, which is a shape formed by rotating the curve \( y = \frac{1}{x} \) around the x-axis. The horn extends infinitely to the right and flares outward from the point \( x = 1 \) as it approaches the y-axis, creating an infinitely long, yet finite-volume shape.**Mathematical Explanation:**\[ S = \left( \, \int_1^{\infty} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \right) \]\[ = 2\pi \int_1^{\infty} \left( \frac{1}{x} \right) \sqrt{1 + \frac{1}{x^4}} \, dx \]\[ = 2\pi \int_1^{\infty} \frac{\sqrt{x^4 + 1}}{x^3} \, dx \]**Note:**Rather than trying to evaluate this integral, note the following:\[ \sqrt{x^4 + 1} \geq \sqrt{x^4} = x^2 \quad \text{for } x > 0 \]This inequality indicates that the integrand behaves in a manner that leads to an infinite surface area as \( x \) approaches infinity, while the volume remains finite. ### Understanding Solid of Revolution: The Gabriel's HornThe image illustrates a solid of revolution known as Gabriel's Horn, formed by rotating the curve \( y = \frac{1}{x} \) around the x-axis for \( x \geq 1 \).#### Surface Area CalculationTo determine the surface area \( S \) of this solid, we use the formula:\[S = 2\pi \int_{1}^{\infty} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]After substituting \( y = \frac{1}{x} \) and simplifying, we have:\[S = 2\pi \int_{1}^{\infty} \left( \sqrt{1 + \frac{1}{x^4}} \right) \, dx \]This transforms into:\[S = 2\pi \int_{1}^{\infty} \frac{\sqrt{x^4 + 1}}{x^3} \, dx \]#### Integral ApproximationRather than evaluating this complicated integral directly, we approximate:\[\sqrt{x^4 + 1} \approx x^2 \quad \text{for} \quad x > 0\]This approximation leads to:\[S \approx 2\pi \int_{1}^{\infty} \frac{x^2}{x^3} \, dx = 2\pi \int_{1}^{\infty} \frac{1}{x} \, dx \]#### ConclusionThe integral:\[\int_{1}^{\infty} \frac{1}{x} \, dx\]is known to be divergent, implying that the surface area \( S \) of Gabriel's Horn is infinite. Despite its finite volume, the infinite surface area poses an interesting paradox.

1 S = 1 + dx y dx = 2n ---Select--- v 1 + 1 dx x + 1 00 = 2n dx ---Select--- v Rather than trying to evaluate this integral, note the following. V x4 + 1 ? v Vx4 = x for x > 0 Thus, if the area is finite, we have the following. x4 + 1 o y2 dx = 2n S = 2n dx ? v 2n dx /1 Ji But we know that this integral ---Select--- v so the area S is infinite. 118