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 (i) about the x-axis.

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

Set up the volume integral using the disk method because the region is revolved around the x-axis. The radius of each disk is the distance from the x-axis to the curve, which is \(r(x) = \frac{1}{x^2}\).

Write the volume integral as \(V = \pi \int_{1}^{\infty} \left(r(x)\right)^2 \, dx = \pi \int_{1}^{\infty} \left(\frac{1}{x^2}\right)^2 \, dx\).

Simplify the integrand: \(\left(\frac{1}{x^2}\right)^2 = \frac{1}{x^4}\), so the integral becomes \(V = \pi \int_{1}^{\infty} \frac{1}{x^4} \, dx\).

Evaluate the improper integral by finding the antiderivative of \(x^{-4}\), then take the limit as the upper bound approaches infinity to determine the volume.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integrals for Area and Volume

Definite integrals calculate the accumulation of quantities, such as area under a curve or volume of solids. In volume problems, integrals sum infinitesimal cross-sectional areas along an axis to find total volume. Setting correct limits and integrand expressions is essential.

Method of Disks/Washers for Volumes of Revolution

The disk/washer method finds volumes by slicing the solid perpendicular to the axis of revolution. Each slice forms a disk or washer whose area is integrated along the axis. For revolution about the x-axis, the radius is the function value, and volume is π∫[radius]^2 dx.

Understanding the Region and Boundaries

Identifying the region bounded by y = 1/x², y = 0, and x = 1 in the first quadrant is crucial. This defines the limits of integration and the shape being revolved. Recognizing that the region extends from x=1 to infinity helps set proper integral bounds for volume calculation.

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 (i) about the x-axis.