Lecture_6 - CALCULUS II Spring 2009 LECTURE 6 This lecture...

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Unformatted text preview: CALCULUS II Spring 2009 LECTURE 6 This lecture provides a model for areas of surfaces of revolution. It contains several examples in which areas are computed. 1 Recall We recall from Lecture 5: • a definition of arc length that is independent of a particular curve model • the integral R b a p 1 + ( y ) 2 dx can be used for arc length evaluation of smooth real-valued functions • examples of arc length calculations. End of Recall 2 Surface Area In a previous lecture, we were able to define the arc length of a planar curve modeled by y ( x ) , a ≤ x ≤ b . Specifically, we approximated the curve by a piecewise linear function, computed the length of the piecewise linear func- tion and then used the least upper bound of such lengths as the length of the curve. Our success there hinged on the fact that we had a standard, namely the length of a planar chord. For a surface defined by a real-valued function of two real variables ( z = f ( x,y )) there is an analogous process. It is possible to triangulate the sur- face, sum the areas of the triangular facets and then find the sum’s least upper bound over all triangulations. However, that study transcends the mathematics of this treatise. This leaves us in the awkward position of wanting to compute surface area without having an underlying definition of surface area. Our work-around is to use human intuition and define a notion of surface area for a special class of surfaces. For surfaces in this class our definition of surface area is based on a limiting process that in turn leads to an integral. We remain mindful of the fact that our surface area calculations are based on our special definition–that a surface in this class may also fall in another class for which another definition of area is provided. (In such an event we have a potential for conflict.) In order to get started, we mentally draw the curve y ( x ) = √ x, 1 ≤ x ≤ 2 on an x,y-plane. Having done that we imagine the 3-d surface obtained by rotating this curve about the x-axis. We seek to model our intuitive notion of the area of this surface. (For example, one needs a notion of surface area in order to buy the correct amount of paint needed to paint such a surface. One needs a notion of surface area in order to determine the heat radiation characteristics of such a surface.) We continue to think (behind the scenes) in terms of y ( x ) = √ x, 1 ≤ x ≤ 2, but consider a general non-negative real-valued differentiable function y ( x ) , a ≤ x ≤ b . One partitions the interval [ a,b ] by a = x < x 1 < ··· < 3 x n- 1 < x n = b . One mentally draws the (small) arc of this curve between the two points ( x i- 1 ,y i- 1 ) and ( x i ,y i ) and generates a (small) surface by rotating this arc around the x-axis. (We let y i = y ( x i ) , ≤ i ≤ n .) This brings us to the crucial question in our modeling process: How can one...
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This note was uploaded on 02/09/2009 for the course MATH 1020 taught by Professor Ecker during the Spring '08 term at Rensselaer Polytechnic Institute.

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Lecture_6 - CALCULUS II Spring 2009 LECTURE 6 This lecture...

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