121616949-math.239 - lim n →∞ n-1 X i =0 2 πf(¯ x i p...

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9.10 Surface Area 225 curve is rotated around the x -axis, it forms a frustum of a cone. The area is 2 πrh = 2 πf x i + x i +1 2 p 1 + ( f ( t i )) 2 Δ x. Note that f (( x i + x i +1 ) / 2) is the average of the two radii, and p 1 + ( f ( t i )) 2 Δ x is the length of the line segment, as we found in the previous section. If we abbreviate f (( x i + x i +1 ) / 2) = f x i ), the approximation for the surface area is n - 1 X i =0 2 πf x i ) p 1 + ( f ( t i )) 2 Δ x. This is not quite the sort of sum we have seen before, as it contains two different values in the interval [ x i , x i +1 ], namely ¯ x i and t i . Nevertheless, using more advanced techniques
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Unformatted text preview: lim n →∞ n-1 X i =0 2 πf (¯ x i ) p 1 + ( f ± ( t i )) 2 Δ x = Z b a 2 πf ( x ) p 1 + ( f ± ( x )) 2 dx is the surface area we seek. x i ¯ x i x i +1 ............................................... ( x i , f ( x i )) ( x i +1 , f ( x i +1 )) Figure 9.24 One subinterval. EXAMPLE 9.38 We compute the surface area of a sphere of radius r . The sphere can be obtained by rotating the graph of f ( x ) = p r 2-x 2 about the x-axis. The derivative...
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