Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.

검증된 단계별 안내

Identify the region described in Exercise 45. Since the problem references a previous exercise, first recall or write down the functions and the interval that define the region to be revolved around the x-axis.

Set up the volume integral using the disk or washer method. Because the solid is generated by revolving around the x-axis, the volume \(V\) can be expressed as \(V = \pi \int_a^b [R(x)]^2 \, dx\), where \(R(x)\) is the radius of the cross-sectional disk at position \(x\).

Determine the radius function \(R(x)\). This radius is typically the distance from the x-axis to the curve defining the boundary of the region. If there are two curves, use the difference of their values to find the outer and inner radii for the washer method.

Set the limits of integration \(a\) and \(b\) based on the interval over which the region extends along the x-axis.

Write the integral explicitly with the squared radius function and limits, then prepare to evaluate the integral to find the volume. (Do not compute the integral yet, just set it up.)

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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 volume is typically calculated using integral calculus, where the cross-sectional area perpendicular to the axis of rotation is integrated along the axis.

Disk and Washer Methods

These are techniques to compute volumes of solids of revolution. The disk method applies when the solid has no hole (solid disks), while the washer method is used when there is a hollow center, involving subtracting the inner radius area from the outer radius area in the integral.

Setting up the Integral with Proper Limits

To find the volume, you must correctly identify the bounds of integration along the axis of rotation and express the radius (or radii) of the disks or washers as functions of x (or y). Accurate limits and radius expressions ensure the integral represents the volume precisely.