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Unformatted text preview: 6.2: Volumes (Dated: October 1, 2011) A (right) cylinder is a solid bounded by two plane con- gruent regions, say B 1 and B 2 , lying in parallel planes. It is customary to call one of these planes, say B 1 , the base of the cylinder. The height of the cylinder is the length of any of the line segments that are perpendicular to the base and join B 1 to B 2 . Let A be the area of B 1 . Then the volume of the cylinder is given by V = Ah . The volume of a solid S that isnt a cylinder can be ap- proximated by a Riemann sum whose summands are vol- umes of constituent cylinders. Lets clarify this. Suppose the x-coordinates of the points within S extend from x = a to x = b . Let P x be a plane perpendicular to the x-axis and passing through the point x , where a x b . The plane region obtained by intersecting S with P x is called the cross-section of S at x and has an area A ( x ). (In general, the cross-sectional area A ( x ) varies as x increases from a to b .) Now lets slice the solid S into n slabs of equal width x = b- a n by using the planes P a = P x , P x 1 , P x 2 , , P x n- 1 , and P b = P x n , where x = a , x 1 = a + x , , x n- 1 = a + ( n- 1) x , x n = b . This way the i th slab S i is the part of S that lies between P x i- 1 and P x i . Choosing a sample point x * i in [ x i- 1 , x i ], one can approximate S...
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