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Volumes of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.

Slicing the solid into thin slabs

The formula of volume of Solid The volume V of the solid should be given approximately by the Riemann sum

Solids of Revolution 1. The Method of Disks 2. The Method of Shells

Example Let R be the regions bounded by the graphs of and Compute the volume of the solid formed by revolving R about y-axis

Revolving about y-axis

The Method of Disks

The Method of Disks (2)

Revolving about x-axis

Revolving about x-axis (animate)

The Solid by revolving about x-axis

Example Let R be the regions bounded by the graphs of and Compute the volume of the solid formed by revolving R about y-axis

Example(continued)

Example Let R be the regions bounded by the graphs of and Compute the volume of the solid formed by revolving R about x-axis

Example(continued)