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Unformatted text preview: Math 113 Lecture #18 Â§ 8.2: Area of a Surface of Revolution Defining the Area of a Surface of Revolution. Revolving a curve about a line forms a surface of revolution. We can think of a surface of revolution as the lateral boundary of the solid of revolution obtained by revolution that curve about the line. A simple example of a surface of revolution is the right cylinder of base radius r and lateral height h . 1.0 0.6 0.8 y 0.6 0.4 0.4 0.0 0.2 0.2 0.0 x 1.2 0.8 1.4 1.0 By cutting this right cylinder along the lateral side, we can unfold the right cylinder into a rectangle of base 2 Ï€r and height h , and thus is area is A = 2 Ï€rh. A less obvious example is that of a circular cone with base radius r and slant height l . y 1.8 1.4 0.6 0.2 x 1.0 0.8 0.4 0.6 0.0 0.2 1.6 1.0 0.8 0.0 2.0 1.2 0.4 By cutting this circular cone along a vertical from the base to the point, we can unfold the circular cone into a sector of a circle with radius r and angle Î¸ = 2 Ï€r/l . Remember the definition of angle: the length of the arc of the sector of the circle divided by the radius. Recall that the area of a sector of radius l and angle Î¸ is (1 / 2) l 2 Î¸ . Think of the area of a circle of radius l , which is Ï€l 2 , i.e., the angle Î¸ is 2 Ï€ ....
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This note was uploaded on 10/29/2010 for the course ENGR 131 taught by Professor Purzer during the Spring '10 term at Purdue.
- Spring '10