Finding volume
The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

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Identify the region bounded by the coordinate axes and the curve \(y = -\ln x\) in the first quadrant. Since \(y = -\ln x\), and we are in the first quadrant, \(x\) ranges from 0 to 1 because \(\ln x\) is negative for \(0 < x < 1\) and \(y\) is positive there.

Set up the volume integral using the method of disks/washers. When revolving around the x-axis, the volume element is \(dV = \pi [f(x)]^2 dx\), where \(f(x)\) is the radius of the disk. Here, the radius is \(y = -\ln x\).

Write the volume integral as \(V = \pi \int_0^1 (-\ln x)^2 \, dx\). This integral represents the volume of the solid generated by revolving the region around the x-axis.

To evaluate the integral \(\int_0^1 (-\ln x)^2 \, dx\), consider using the substitution \(t = -\ln x\), which implies \(x = e^{-t}\) and \(dx = -e^{-t} dt\). Change the limits accordingly: when \(x=0^+\), \(t \to \infty\), and when \(x=1\), \(t=0\).

Rewrite the integral in terms of \(t\) and evaluate it using integration techniques for exponential and polynomial functions. This will give the volume of the solid.

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주요 개념

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

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The disk or washer method is commonly used, where the volume is computed by integrating the cross-sectional area perpendicular to the axis of rotation.

Natural Logarithm Function and Its Properties

The function y = -ln(x) is defined for x > 0 and decreases as x increases. Understanding its behavior and domain is essential to correctly set up the limits of integration and the shape of the region bounded by the axes and the curve.

Definite Integration and Limits of Integration

Definite integrals calculate the accumulated area or volume between specified bounds. Identifying the correct limits, especially for improper integrals when the region extends to infinity, is crucial for accurately computing the volume.

Finding volumeThe infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.