lect116_31rev_f11

lect116_31rev_f11 - Thursday, November 24 Lecture 31 :...

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Thursday, November 24 Lecture 31 : Volumes of solids of revolution: cylindrical shells method. (Refers to Section 7.3 in your text) Students who have mastered the content of this lecture know : what computing the volume of solid by cylindrical shells means, Students who have practiced the techniques presented in this lecture will be able to : Compute the volume of solids by cylindrical shells around the x-axis or the y-axis. 31.1 Introduction To find the volume of a solid by the " cross-sections method " one must be able to easily find a formula for A ( x ) or A ( y ) (the cross-section areas at x or y ). - For some solids this is difficult to do. - There is another method for calculating the volume of certain solids of revolution. It is called the cylindrical shells method . We use it when it simplifies the computation of the volume. 31.1.1 Definition A cylindrical shell is a region bounded by two concentric circular cylinders of constant height h . and constant thickness x . 31.1.2 Suppose the radius of the inner cylinder is r 1 , and the radius of the outer shell is r 2 . Then the volume of the cylindrical shell is given by the expression 31.2 The principle behind the cylindrical shell method . - Suppose we are given a solid of revolution formed by rotating the region under y = f ( x) over [ a, b ] around the y -axis. - 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 . - Above the i th interval [ x i 1 , x i ] raise a rectangle of height f ( x i ) (a right endpoint column). Then revolve this thin rectangle about the y -axis to obtain a solid of revolution in the form of a cylindrical shell. Let denote the volume of this i th cylindrical . We see that V is a function of x i .

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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_31rev_f11 - Thursday, November 24 Lecture 31 :...

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