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Unformatted text preview: Lecture 8: MATH 138-W12-003 January 20, 2012 I) Volumes by disks, Section 6.2 Now that weve established a multitude of techniques of integration we are going to learn how to apply them to the very physical problem of computing volumes. Since we only know calculus in one dimension we cannot do any arbitrary volumes but only those with a rotational symmetry. The interested student who wants to learn the more general technique of computing volumes is directed to MATH 237 (Calculus 3). Regular Cylinders We define a cylinder (or a right cylinder) to be a cylinder bounded by a plane region, say B 1 , called the base and a congruent region B 2 in a parallel plane to the base. The cylinder consists of all points that are perpendicular to the base and join B 1 to B 2 . It is a simple geometric property that if the area of the base is A and the height of the cylinder is h then the volume inside the cylinder is, V = Ah (volume of right cylinder) . If the base is a square of area s 2 then the volume of s 2 h (volume of rectangular box) . and if the base is an ellipse of area ab then the volume is abh (volume of elliptical cylinder) . Figure 1: This is an example of a regular circular cylinder....
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