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Unformatted text preview: 8.2: Area of a Surface of Revolution (Dated: November 8, 2011) THE SURFACE AREA OF A FRUSTUM OF A CIRCULAR CONE Consider a circular cone with base radius r and slant height l . It can be flattened to form a sector of a circle with radius l and central angle θ = 2 π r 2 π l · 2 π = 2 π r l . It follows that the surface area of the cone is given by A = θ 2 π · π l 2 = π rl . Consider instead the frustum of a cone with slant height l and upper and lower radii r 1 and r 2 respectively. If the slant height of the cone is l 1 + l , then the surface area of the frus tum is given by A = π r 2 ( l 1 + l ) π r 1 l 1 = π [( r 2 r 1 ) l 1 + r 2 l ]. On the other hand, from similar triangles one has l 1 r 1 = l 1 + l r 2 , from which it follows that ( r 2 r 1 ) l 1 = r 1 l . Hence the area of the frustum is also given by A = π ( r 1 + r 1 ) l = 2 π rl , where r = r 1 + r 2 2 is the average radius of the frustum. THE AREA OF A SURFACE OF REVOLUTION A surface of revolution can be obtained by rotating a curve y = f ( x ), a ≤ x ≤ b , about the xaxis, where f is positive and smooth. To calculate the area S of the sur...
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This note was uploaded on 11/11/2011 for the course MATH 152.01 taught by Professor Geline during the Fall '09 term at Ohio State.
 Fall '09
 GELINE
 Calculus, Geometry, Cone

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