lect116_30rev_f11

lect116_30rev_f11 - Tuesday, November 22 Lecture 30 :...

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Tuesday, November 22 Lecture 30 : Volumes of a solid of revolution: Cross- section (slicing) method (Refers to Section 7.2 in your text) Students who have mastered the content of this lecture know : what computing the volume of solid by cross-section means, about solids of revolution, the formula for computing the volume of a solid of revolution by cross-sections. Students who have practiced the techniques presented in this lecture will be able to : Compute the volume of solids by using cross-sections in cases where the solid is a solid of revolution and cases where it is not. 30.1 Introduction In this lecture we study methods for determining the volume of a few simple solids for which we can obtain a formula for its cross-sections. 30.1.1 Definition Calculating the volume by the method of cross sections (referred to as slicing in the text) is done by determining the area A ( x ) of vertical cross sections of a solid at x , or the area A ( y ) of a horizontal cross section of the solid at y . 30.2 The principle behind the cross section method . - Suppose we are given a solid S lying alongside, " skewed " by the x -axis, whose extremities are given by the planes x = a and y = b , in 3-space. - Let us subdivide the interval [ a, b ] into n equal subintervals a = x 0 , x 1 , x 2 , ..., x n 1 , x n = b, each of length x = ( b a ) / n . - Let A ( x i ) denote the area of a cross section of the solid over the interval [ x i 1 , x i ] i.e., the area of the region formed by the intersection of the plane x = x i in 3-space with the solid S . The symbol x i represents the right endpoint of the i th interval x . - Thus A ( x 1 ), A ( x 2 ), ..., A ( x n ) represent a sequence of areas of cross sections of the solid S at x 1 , x 2 , ..., x n 1 , x n .
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- Then A ( x 1 ) x , A ( x 2 ) x , ..., A ( x n ) x represent a sequence of volumes (base area times height of n solids lying on their sides: The A ( x i ) can be seen as the area of the base of the i th solid and x as its height. - Then the Σ i = 1 to n A ( x i ) x expression denotes the sum of these n volumes. -
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This note was uploaded on 12/28/2011 for the course MATH 116 taught by Professor Robertandre during the Spring '11 term at Waterloo.

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lect116_30rev_f11 - Tuesday, November 22 Lecture 30 :...

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