42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
42. The region bounded by f(x) = ln(x), y = 1, and the coordinate axes is revolved about the x-axis.

검증된 단계별 안내

First, identify the region bounded by the curves: \(y = \ln(x)\), \(y = 1\), the \(x\)-axis (\(y=0\)), and the \(y\)-axis (\(x=0\)). Since \(\ln(x)\) is defined for \(x > 0\), the region lies between \(x=1\) (where \(\ln(1)=0\)) and \(x=e\) (where \(\ln(e)=1\)).

Set up the volume integral using the method of washers (disks with holes) since the solid is generated by revolving the region around the \(x\)-axis. The volume element is \(\pi \left(R^2 - r^2\right) dx\), where \(R\) is the outer radius and \(r\) is the inner radius of the washer at position \(x\).

Determine the outer and inner radii of the washers. The outer radius corresponds to the line \(y=1\), so \(R = 1\). The inner radius corresponds to the curve \(y = \ln(x)\), so \(r = \ln(x)\).

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

To find the volume, evaluate the integral \(\int_{1}^{e} \left(1 - (\ln(x))^2\right) dx\). This may require integration by parts or substitution for the \(\int (\ln(x))^2 dx\) term.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes of slices or shells.

Disk/Washer Method

The disk/washer method calculates volume by slicing the solid perpendicular to the axis of rotation, forming circular disks or washers. The volume is found by integrating the area of these cross-sections along the axis of revolution.

Natural Logarithm Function and Its Graph

Understanding the function f(x) = ln(x) is essential, including its domain (x > 0), range, and shape. Knowing how it interacts with the lines y = 1 and the coordinate axes helps define the bounded region to be revolved.

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.42. The region bounded by f(x) = ln(x), y = 1, and the coordinate axes is revolved about the x-axis.