6.3 Volumes of Revolution - Volumes of Revolution Discs and...

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Volumes of Revolution: Discs and Washers
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Volumes of Revolution Volumes of revolutions are formed by rotating a planar region about a given axis. Consider, for instance, the region bounded by the function y = f ( x ) and the vertical lines x = a and x =b. When the region is rotated about the x- axis we obtain the following solid:
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Volumes: Discs We start by partitioning the interval [ a , b ] into n subintervals of equal width and creating the corresponding rectangles using, for instance, the left end points. In this example, n = 8. When the region is rotated about the x-axis, each rectangle forms a slab (in this example the slab is a disk). The volume of the solid is roughly the sum of the volumes of the slabs. 1 n i i V A x x thickness of the slab x ( ) area of the base of the slab i A x area of cross section perpendicular to the axis of rotation
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Volumes: Discs The height of the rectangle, , equals the radius R of the disk. And the total volume of the solid: 2 1 ( ) n i i V f x x Each cross sectional area is that of a circle: 2 2 ( ) ( ) i i A x R f x We now look at a sample rectangle and the corresponding disk. ( ) i f x
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Volume: Discs 1 ( ) lim n b i a n i V A x x A x dx  2 1 ( ) lim n b i a n i V A x x f x dx
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  • Fall '08
  • SURGENT
  • Region, volumes

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