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Chapter 6 | Applications of Integration
Figure 6.22 (a) A thin rectangle in the region between two curves. (b) A representative disk formed by revolving the rectangle about the x -axis. (c) The region between the curves over the given interval. (d) The resulting solid of revolution.
The cross-sectional area, then, is the area of the outer circle less the area of the inner circle. In this case, A ( x ) = π ( x ) 2 − π (1) 2 = π ( x −1). Then the volume of the solid is V = ∫ a b A ( x ) dx = ∫ 1 4 π ( x −1) dx = π ⎡ ⎣ x 2 2 − x ⎤ ⎦ | 1 4 = 9 2 π units 3 . Generalizing this process gives the washer method . Rule: The Washer Method Suppose f ( x ) and g ( x ) are continuous, nonnegative functions such that f ( x ) ≥ g ( x ) over ⎡ ⎣ a , b ⎤ ⎦ . Let R denote the region bounded above by the graph of f ( x ), below by the graph of g ( x ), on the left by the line x = a , and on
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