Applications of Integration

(cod: P-79-52-8) Consider the surface of revolution (unbounded) obtained by rotating the graph of the function \[f(x) = \dfrac{1}{x}, \, \, \, x \in [1,+\infty)\] around the x-axis. This surface is called the Gabriel's trumpet.

Let \(\mathcal{R}\) be the unbounded region located below the graph of \(f\) and above the \(x\) axis. Denote by \(W\) the solid obtained by rotating the region \(\mathcal{R}\) around the x-axis, that is, \(W\) is the solid bounded by Gabriel's trumpet.

Using improper integration, we can state that:

The surface area of the trumpet is twice the volume enclosed by it (i.e., the volume of \(W\)).

Both the surface area of the trumpet and the volume enclosed by it (i.e., the volume of \(W\)) are finite.

Both the surface area of the trumpet and the volume enclosed by it (i.e., the volume of \(W\)) are infinite.

The surface area of the trumpet is infinite, while the volume enclosed by it (i.e., the volume of \(W\)) is finite.

The surface area of the trumpet is less than the volume enclosed by it (i.e., the volume of \(W\)).