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lec31 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture 31 18.01 Fall 2006 Lecture 31: Parametric Equations, Arclength, Surface Area Arclength, continued Example 1. Consider this parametric equation: x = t 2 y = t 3 for ≤ t ≤ 1 x 3 = ( t 2 ) 3 = t 6 ; y 2 = ( t 3 ) 2 = t 6 = ⇒ x 3 = y 2 = ⇒ y = x 2 / 3 ≤ x ≤ 1 ds dy dx ds dy dx Figure 1: Infinitesimal Arclength. ( ds ) 2 = ( dx ) 2 + ( dy ) 2 ( ds ) 2 = (2 tdt ) 2 + (3 t 2 dt ) 2 = (4 t 2 + 9 t 4 )( dt ) 2 ( dx ) 2 ( dy ) 2 t =1 1 1 Length = ds = 4 t 2 + 9 t 4 dt = t 4 + 9 t 2 dt t =0 1 = (4 + 9 t 2 ) 3 / 2 = 1 (13 3 / 2 − 4 3 / 2 ) 27 27 Even if you can’t evaluate the integral analytically, you can always use numerical methods. 1 Lecture 31 18.01 Fall 2006 Surface Area (surfaces of revolution) y ds a b y x Figure 2: Calculating surface area ds (the infinitesimal curve length in Figure 2) is revolved a distance...
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lec31 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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