Engineering Calculus Notes 115

Engineering Calculus Notes 115 - orienting the base and...

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1.7. APPLICATIONS OF CROSS PRODUCTS 103 generalize this, replacing the discs with horizontal copies of any plane region, and allowing the two copies to not be directly above one another (so the line segments joining their boundaries, while parallel to each other, need not be perpendicular to the two regions). Another way to say this is to deFne a (solid) cylinder on a given base (which is some region in a plane) to be formed by parallel line segments of equal length emanating from all points of the base (±igure 1.52 ). We will refer to a vector −→ v representing these segments as a generator for the cylinder. B −→ v h ±igure 1.52: Cylinder with base B , generator −→ v , height h Using Cavalieri’s principle ( Calculus Deconstructed , p. 365) it is fairly easy to see that the volume of a cylinder is the area of its base times its height (the perpendicular distance between the two planes containing the endpoints of the generating segments). Up to sign, this is given by
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Unformatted text preview: orienting the base and taking the dot product of the generator with the oriented area of the base V = v A ( B ) . We can think of this dot product as the signed volume of the oriented cylinder, where the orientation of the cylinder is given by the direction of the generator together with the orientation of the base. The signed volume is positive ( resp . negative) if v points toward the side of the base from which its orientation appears counterclockwise ( resp . clockwise)in other words, the orientation of the cylinder is positive if these data obey the right-hand rule. We will denote the signed volume of a cylinder C by V ( C ). A cylinder whose base is a parallelogram is called a parallelepiped : this has three quartets of parallel edges , which in pairs bound three pairs of...
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