89. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / x², y = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.

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Identify the region to be revolved: it is bounded by the curve \(y = \frac{1}{x^{2}}\), the line \(y = 0\), and the vertical line \(x = 1\), all in the first quadrant.

Since the region is revolved about the y-axis, consider using the method of cylindrical shells. The shell radius is the distance from the y-axis, which is \(x\), and the shell height is the vertical distance between the curve and the x-axis, which is \(\frac{1}{x^{2}}\).

Set up the volume integral using the shell method formula: \(V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx\). Here, \(a = 1\) and \(b \to \infty\) because the region extends infinitely to the right.

Write the integral explicitly: \(V = 2\pi \int_{1}^{\infty} x \cdot \frac{1}{x^{2}} \, dx = 2\pi \int_{1}^{\infty} \frac{1}{x} \, dx\).

Evaluate the improper integral by taking the limit as the upper bound approaches infinity, and analyze whether the volume converges or diverges.

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주요 개념

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Setting up the Region for Integration

Understanding the boundaries of the region is essential. Here, the region is in the first quadrant bounded by y = 1/x², y = 0, and x = 1. Identifying these limits helps determine the interval and shape of the area to be revolved.

Method of Cylindrical Shells for Volume

When revolving a region around the y-axis, the cylindrical shells method is often used. It involves integrating the volume of thin cylindrical shells with radius x, height given by the function, and thickness dx, which simplifies volume calculation.

Improper Integrals and Convergence

Since the region extends infinitely in the y-direction, the volume integral is improper. Evaluating whether the integral converges (has a finite value) is crucial to ensure the volume is well-defined and to compute it correctly.

89. Consider the infinite region in the first quadrant bounded by the graphs ofy = 1 / x², y = 0, and x = 1.b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.