102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.
104. About the line y = 1

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Identify the region R bounded by the curve \(y = \ln(x)\) and the x-axis on the interval \([1, e]\). This means the region lies between \(y = 0\) and \(y = \ln(x)\) for \(x\) from 1 to \(e\).

Since the solid is generated by revolving the region around the line \(y = 1\), consider the distance from the curve and the x-axis to this line. The radius of a typical disk or washer will be the vertical distance from \(y = 1\) to the curve or axis.

Set up the volume integral using the washer method. The outer radius \(R_{outer}\) is the distance from \(y = 1\) to the x-axis (\(y=0\)), which is \(1 - 0 = 1\). The inner radius \(R_{inner}\) is the distance from \(y = 1\) to the curve \(y = \ln(x)\), which is \(1 - \ln(x)\).

Write the volume integral as \(V = \pi \int_1^{e} \left(R_{outer}^2 - R_{inner}^2\right) \, dx = \pi \int_1^{e} \left(1^2 - (1 - \ln(x))^2\right) \, dx\).

Expand the integrand and simplify it before integrating. Then, integrate term-by-term with respect to \(x\) over the interval \([1, e]\) to find the volume.

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주요 개념

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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 a line. The volume is typically calculated using integral methods such as the disk, washer, or shell method, depending on the axis of rotation and the shape of the region.

Washer Method

The washer method is used when the solid has a hollow center, created by revolving a region around a line that does not coincide with the axis of the region. It involves subtracting the volume of the inner radius from the outer radius, integrating π(outer radius² - inner radius²) over the interval.

Natural Logarithm Function and Its Graph

Understanding the function y = ln(x) is essential, including its behavior on the interval [1, e]. The curve lies above the x-axis here, and knowing its shape helps determine the radii for the washers when revolving around the line y = 1.

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.104. About the line y = 1