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Unformatted text preview: Chapter 5 Applications of the definite integral to calculating volume and length In this chapter we consider applications of the definite integral to calculating geometric quantities such as volumes. The idea will be to dissect the three dimensional objects into pieces that resemble disks or shells, whose volumes we can approximate with simple formulae. The volume of the entire object is obtained by summing up volumes of a stack of disks or a set of embedded shells, and considering the limit as the thickness of the dissection cuts gets thinner. In Figure 5.1 we first remind the reader of the volumes of some of the geometric shapes that will be used as elementary pieces into which our shapes will be carved. Recall that, from earlier discussion, we have r h r disk shell Figure 5.1: The volumes of these simple 3D shapes are given by simple formulae. We use them as basic elements in computing more complicated volumes. 1. The volume of a cylinder of height h having circular base of radius r , is V cylinder = r 2 h v.2005.1  January 5, 2009 1 Math 103 Notes Chapter 5 2. The volume of a circular disk of thickness , and radius r (shown in Figure 5.1 a), as a special case of the above, is V disk = r 2 . 3. The volume of a cylindrical shell of height h , with circular radius r and small thickness (shown in Figure 5.1 b) is V shell = 2 rh. (This approximation holds for << r .) 5.1 Solids of revolution In our first approach to volumes, we will restrict attention to solids of revolution , i.e. volumes enclosed by some curve (described by a function such as y = f ( x )), when it is rotated about one of the axes. In Figure 5.2(a) we show one such curve, and the surface it forms when it is revolved about the x axis. We note that if this surface is cut into slices along the x axis, the crosssections look like circles. (The circle will have a radius that depends on the position of the cut.) In Figure 5.2(b) we show how a set of disks of various radii can approximately represent the shape of interest. The total (a) (b) r x y=f(x) Figure 5.2: (a) A solid of revolution, showing dissection into slices along its axis of rotation. (b) The same volume is approximated by a set of disks. Each disk has some radius r (that varies along the length of the object) and thickness x . Note that the thickness is in the direction of the x axis: this will remind us that we integrate with respect to x . volume of these disks is not the same, clearly, as the volume of the object, since some of these stick out beyond the surface. However, if we make the thickness of these disks very small, we will get a v.2005.1  January 5, 2009 2 Math 103 Notes Chapter 5 good approximation of the desired volume. In the limit, as the thickness becomes infinitesimal, we arrive at the true volume....
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This note was uploaded on 02/17/2012 for the course MTH 133 taught by Professor Staff during the Fall '08 term at Michigan State University.
 Fall '08
 STAFF
 Calculus, The American

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