65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.

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Identify the function and the interval: The function is given as \(f(x) = (x^2 + 1)^{-\frac{1}{2}}\), and the interval is \([2, \infty)\).

Set up the volume integral using the disk method since the region is revolved about the x-axis. The volume \(V\) is given by the integral \(V = \pi \int_{2}^{\infty} [f(x)]^2 \, dx\).

Substitute the function into the integral: \(V = \pi \int_{2}^{\infty} \left((x^2 + 1)^{-\frac{1}{2}}\right)^2 \, dx = \pi \int_{2}^{\infty} \frac{1}{x^2 + 1} \, dx\).

Evaluate the improper integral by finding the antiderivative of \(\frac{1}{x^2 + 1}\), which is \(\arctan(x)\), and then apply the limits from 2 to infinity.

Determine if the volume converges by calculating the limit \(\lim_{b \to \infty} \pi [\arctan(b) - \arctan(2)]\). If this limit exists and is finite, the volume exists; otherwise, it does not.

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Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method, which use integrals to sum infinitesimal volumes. Understanding how to set up these integrals based on the axis of rotation and the given function is essential.

Improper Integrals and Convergence

When the region extends to infinity, the volume integral becomes an improper integral. Determining whether the volume exists requires analyzing the convergence of this integral. Techniques include evaluating limits and applying comparison tests to ensure the integral yields a finite value.

Function Behavior and Boundedness

Understanding the behavior of the function f(x) = (x^2 + 1)^(-1/2) on [2, ∞) is crucial. Since it decreases and approaches zero as x approaches infinity, this affects the shape and size of the solid. Recognizing how the function's decay influences the volume helps in setting up and evaluating the integral correctly.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.