Let R be the region bounded by the following curves. Use the...

Let R be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when $R$ is revolved about the $x$ -axis.
$y=2 x, y=0, x=3$ (Verify that your answer agrees with the volume formula for a cone.)

Key Concepts

Disk Method

The disk method is a technique in calculus used to compute the volume of a solid of revolution. This is done by slicing the solid perpendicular to the axis of rotation, forming disk-shaped slices whose volumes are approximated by ? times the square of the radius integrated along the axis. This method is particularly useful when rotating a function or region about a coordinate axis and then summing the volumes of the disks to obtain the total volume.

Definite Integration

Definite integration is used to calculate exact area or volume by summing up an infinite number of infinitesimally small slices. When applying the disk method, the volume is expressed as a definite integral with limits corresponding to the bounds of the region being rotated. This integral formulation allows for the precise computation of the volume of the curved solid created.

Solids of Revolution

Solids of revolution are three-dimensional objects obtained by rotating a two-dimensional region around an axis. Calculus provides systematic methods such as the disk method to compute the volumes of these solids. Understanding how to set up the integral and determine the radius for each infinitesimal disk is essential in calculating the volume of these shapes.

Volume of a Cone

The volume of a cone is given by the formula V = (1/3)?r²h, where r is the radius of the base and h is the height of the cone. This geometric formula provides a quick way to verify the results obtained from calculus methods such as the disk method. Comparing the volume calculated via integration to the formula for a cone helps confirm the correctness of the integration setup and execution.

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