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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 8 - Techniques of Integration문제 8.AAE.18a
8장, 문제 8.AAE.18a

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:
a. the y-axis.

검증된 단계별 안내
1
Identify the region R from Exercise 17. Typically, this region is bounded by given curves or lines. Make sure you clearly understand the boundaries of R before proceeding.
Since the solid is generated by revolving the region R about the y-axis, decide on the method to use for finding the volume. The common methods are the Disk/Washer method or the Shell method. For revolution around the y-axis, the Shell method is often convenient.
Set up the integral using the Shell method. The formula for the volume using cylindrical shells when revolving around the y-axis is: \[ V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx \] Here, the radius is the distance from the y-axis (which is simply \(x\)), and the height is the vertical length of the region at each \(x\).
Determine the limits of integration \(a\) and \(b\) from the domain of the region R along the x-axis. Also, express the height of the shell as a function of \(x\) by subtracting the lower curve from the upper curve within the region.
Write the integral explicitly by substituting the radius and height expressions, then prepare to evaluate the integral to find the volume. Remember, do not compute the final value yet—just set up the integral correctly.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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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 calculating the volume of a 3D solid formed by rotating a 2D region around an axis. The volume is typically found using integral calculus methods such as the disk, washer, or shell methods, depending on the axis of rotation and the shape of the region.
추천 영상:
04:48
Finding Volume Using Disks

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical or horizontal slices of the region around an axis. It is especially useful when revolving around the y-axis and when the region is described in terms of x.
추천 영상:
07:33
Euler's Method

Axis of Rotation (y-axis)

The axis of rotation determines how the region is revolved to form the solid. Revolving around the y-axis means the solid is generated by rotating the region horizontally, which affects the choice of method and the limits of integration.
추천 영상:
06:30
Disk Method Using y-Axis
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