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Chapter 6 | Applications of Integration
(think of an ice cream cone with the pointy end cut off). A piece of a cone like this is called a frustum of a cone. To find the surface area of the band, we need to find the lateral surface area, S , of the frustum (the area of just the slanted outside surface of the frustum, not including the areas of the top or bottom faces). Let r 1 and r 2 be the radii of the wide end and the narrow end of the frustum, respectively, and let l be the slant height of the frustum as shown in the following figure.
Figure 6.42 A frustum of a cone can approximate a small part of surface area.
We know the lateral surface area of a cone is given by
Lateral Surface Area = πrs , where r is the radius of the base of the cone and s is the slant height (see the following figure).
Figure 6.43 The lateral surface area of the cone is given by πrs .
Since a frustum can be thought of as a piece of a cone, the lateral surface area of the frustum is given by the lateral surface area of the whole cone less the lateral surface area of the smaller cone (the pointy tip) that was cut off (see the following figure).
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