7_4_arc_length_surface_revolution

7_4_arc_length_surface_revolution - 7.4 Arc Length and...

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Unformatted text preview: 7.4 Arc Length and Surface of Revolution Study 7.4 p. 483 # 15, 6, 1519, 3945 Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Homework (1 of 14) 7.4 Arc Length and Surface of Revolution arc_length_circle.ggb s = r Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Background (2 of 14) 7.4 Arc Length and Surface of Revolution s = r s = 1 (/2)=1.57 circular_arc.swf Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Background (3 of 14) 7.4 Arc Length and Surface of Revolution Estimate s = 1.44 Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Introduction (4 of 14) 7.4 Arc Length and Surface of Revolution Estimate s = 0.91 + 0.55 = 1.46 Sum of two line segments gets a better estimate. Can increase the number of line segments and let delta x approach 0. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Introduction (5 of 14) 7.4 Arc Length and Surface of Revolution Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: derive arc length formula (6 of 14) 7.4 Arc Length and Surface of Revolution Let y = f(x) be a smooth curve on [a,b]. Then the arc length of f between a and b is Let x = g(y) be a smooth curve on [c,d]. Then the arc length of g between c and d is Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Arc Length Definititon (7 of 14) 7.4 Arc Length and Surface of Revolution p. 483 #4 Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: eg: arc length (8 of 14) 7.4 Arc Length and Surface of Revolution Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: example (9 of 14) 7.4 Arc Length and Surface of Revolution p. 483 #8 Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: eg: arc length (10 of 14) 7.4 Arc Length and Surface of Revolution If we start with an arc length, and rotate it around an axis of revolution, we have a surface of revolution. surface_revolution.swf Surface measurement: Surface Area Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: intro: surface of revolution (11 of 14) 7.4 Arc Length and Surface of Revolution x Let x be small enough so that the surface being rotated is like the surface of a cylinder. Then the surface area is S = 2 r L where L is the arc length. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: intro: surface of revolution (12 of 14) 7.4 Arc Length and Surface of Revolution Let y = f(x) has continuous derivative on [a,b]. The Area S of the surface of revolution formed by revolving the graph of f about a horizontal axis is: where r(x) = distance between and the axis of f revolution. If x=g(y) on [c,d]. The Area S of the surface of revolution formed by revolving the graph of g about a verticle axis is: where r(y) = distance between and the axis of g revolution. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: Definition Surface Area (13 of 14) 7.4 Arc Length and Surface of Revolution Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Title: example (14 of 14) ...
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