Use the disk or washer method to find the volume of the...

Use the disk or washer method to find the volume of the solid of revolution that is formed by revolving the region bounded by the graphs of the given equations about the indicated line or axis.
$y=\frac{1}{x}, x=\frac{1}{2}, x=3, y=0 ; \quad x$ -axis

Key Concepts

Solid of Revolution

A solid of revolution is created when a two-dimensional region is rotated about a line (or axis). This concept is significant in calculus as it allows us to compute the volume and other properties of the three-dimensional object created by the rotation of a plane region.

Disk/Washer Method

The disk method involves slicing a solid perpendicular to the axis of revolution into thin, circular disks, whose volumes are summed via integration. When the region being rotated does not touch the axis, the washer method is used instead; this technique subtracts the volume of a 'hole' in each disk, leading to a formula that involves the difference between the squares of the outer and inner radii.

Determining the Radius Function

In setting up an integral for finding the volume, it is necessary to express the radius of each disk or washer as a function of the variable of integration. This step is crucial because the radius function represents the distance from the axis of revolution to the boundary of the region, dictating the size of each cross-sectional area.

Setting Up Definite Integrals

Once the radius function is determined, the volume is calculated by integrating the cross-sectional area (typically ? times the radius function squared) over the interval defined by the bounds of the region. Establishing the appropriate limits of integration based on where the region is defined ensures the correct accumulation of volume.

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