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Home Applications of the Integral Area of a Surface of Revolution Cylinders and Cones  
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Cylinders and Cones
When a curve in the plane is rotated about the x or yaxis it forms a surface of revolution, as in Figure 6.4.1. Figure 6.4.1 Surfaces of Revolution The simplest surfaces of revolution are the right circular cylinders and cones. We can find their areas without calculus. Figure 6.4.2 Cylinder Figure 6.4.2 shows a right circular cylinder with height h and base of radius r. When the lateral surface is slit vertically and opened up it forms a rectangle with height h and base 2nr. Therefore its area is lateral area of cylinder = 2πhr.
Figure 6.4.3: Cone Figure 6.4.3 shows a right circular cone with slant height l and base of radius r. When the cone is slit vertically and opened up, it forms a circular sector with radius l and arc length s = 2πr. Using the formula A = ½ sl for the area of a sector, we see that the lateral surface of the cone has area lateral area of cone = πrl. Figure 6.4.4: Cone frustum Figure 6.4.4 shows the frustum of a cone with smaller radius r_{1}, larger radius r_{2}, and slant height l. The formula for the area of the lateral surface of a frustum of a cone is lateral area of frustum = π(r_{1} + r_{2})l. This formula is justified as follows. The frustum is formed by removing a cone of radius r_{1} and slant height l_{1} from a cone of radius r_{2} and slant height l_{2}. The frustum therefore has lateral area A = πr_{2}l_{2}  πr_{1}l_{1}. The slant heights are proportional to the radii, so r_{1}l_{2} = r_{2}l_{1}. The slant height l of the frustum is l = l_{2} l_{1}. Using the last two equations,


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