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

Guida verificata passo dopo passo

Identify the function and the interval: The function is given as \(f(x) = \frac{1}{\sqrt{x \ln x}}\) and the interval is \([e, \infty)\).

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 \(V = \pi \int_{e}^{\infty} [f(x)]^2 \, dx\).

Substitute the function into the integral: This gives \(V = \pi \int_{e}^{\infty} \left( \frac{1}{\sqrt{x \ln x}} \right)^2 \, dx = \pi \int_{e}^{\infty} \frac{1}{x \ln x} \, dx\).

Analyze the integral for convergence: To determine if the volume exists, examine the improper integral \(\int_{e}^{\infty} \frac{1}{x \ln x} \, dx\) and check if it converges.

Evaluate or apply a substitution to the integral: Use the substitution \(t = \ln x\), which implies \(dt = \frac{1}{x} dx\), to rewrite the integral in terms of \(t\) and analyze its behavior as \(x \to \infty\).

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Improper Integrals

Improper integrals involve integration over an infinite interval or where the integrand becomes unbounded. In this problem, the interval is [e, ∞), so evaluating the volume requires determining if the integral converges or diverges.

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Percorso guidato

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Improper Integrals: Infinite Intervals

Volume of Solids of Revolution (Disk/Washer Method)

The volume of a solid formed by revolving a region around the x-axis can be found using the disk method, which integrates π[f(x)]² dx over the interval. This method sums the volumes of infinitesimally thin circular disks.

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Finding Volume Using Disks

Behavior of the Function f(x) = 1/√(x ln x)

Understanding the function's behavior, especially its decay rate as x approaches infinity, is crucial to determine if the volume integral converges. Since f(x) involves a logarithmic term in the denominator, its rate of decrease affects the integral's convergence.

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