Volumes
Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

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Identify the region bounded by the curve \(y = \sin x\), the vertical lines \(x = 0\) and \(x = \pi\), and the horizontal line \(y = 2\). This region lies between \(x=0\) and \(x=\pi\) and between \(y=\sin x\) and \(y=2\).

Since the solid is generated by revolving this region about the line \(y = 2\), use the method of washers (disks with holes). The axis of rotation is horizontal and above the curve \(y = \sin x\).

Set up the volume integral using the washer method. The outer radius \(R\) is the distance from the line \(y=2\) to the lower boundary \(y=\sin x\), and the inner radius \(r\) is the distance from \(y=2\) to the line \(y=2\) itself (which is zero). So, \(R = 2 - \sin x\) and \(r = 0\).

Write the volume integral as \(V = \pi \int_0^{\pi} \left(R^2 - r^2\right) \, dx = \pi \int_0^{\pi} (2 - \sin x)^2 \, dx\).

Expand the integrand \((2 - \sin x)^2\), simplify, and then integrate term-by-term over \([0, \pi]\) to find the volume.

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영상 길이:

10m

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

Washer Method

The washer method is used when the solid has a hollow center, formed by revolving a region between two curves around an axis. The volume is found by integrating the difference of the squares of the outer and inner radii, representing the outer and inner disks (washers) perpendicular to the axis of rotation.

Setting up the Integral with Respect to the Axis of Rotation

To correctly compute the volume, one must express the radii of the washers in terms of the distance from the axis of rotation. When revolving around a line other than the x- or y-axis, adjust the radius formulas accordingly, such as using (2 - y) or (y - 2) if revolving around y = 2.

VolumesFind the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.