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

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CALCULUS IISpring 2009LECTURE 6This lecture provides a model for areas of surfaces of revolution.It contains several examples in which areas are computed.1
RecallWe recall from Lecture 5:a definition of arc length that is independent of a particular curvemodelthe integralRbap1 + (y0)2dxcan be used for arc length evaluation ofsmooth real-valued functionsexamples of arc length calculations.End of Recall2
Surface AreaIn a previous lecture, we were able to define the arc length of a planar curvemodeled byy(x), axb. Specifically, we approximated the curve by apiecewise linear function, computed the length of the piecewise linear func-tion and then used the least upper bound of such lengths as the length ofthe 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 leastupper bound over all triangulations.However, that study transcends themathematics of this treatise.This leaves us in the awkward position ofwanting to compute surface area without having an underlying definition ofsurface area.Our work-around is to use human intuition and define a notion of surfacearea for a special class of surfaces. For surfaces in this class our definition ofsurface 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 basedon our special definition–that a surface in this class may also fall in anotherclass for which another definition of area is provided. (In such an event wehave a potential for conflict.)In order to get started, we mentally draw the curvey(x) =x,1x2on anx, y-plane. Having done that we imagine the 3-d surface obtained byrotating this curve about thex-axis. We seek to model our intuitive notionof the area of this surface. (For example, one needs a notion of surface areain 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 radiationcharacteristics of such a surface.)We continue to think (behind the scenes) in terms ofy(x) =x,1x2, but consider a general non-negative real-valued differentiable functiony(x), axb. One partitions the interval [a, b] bya=x0< x1<· · ·<
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xn-1< xn=b. One mentally draws the (small) arc of this curve betweenthe two points (xi-1, yi-1) and (xi, yi) and generates a (small) surface byrotating this arc around thex-axis. (We letyi=y(xi),0in.)This brings us to the crucial question in our modeling process: How can one

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