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Unformatted text preview: osed curve C for which the value of the
line integral
3 ( −3 ) − 3 C is a maximum and determine this maximum value.
S
Let D be the region enclosed by the curve C and it follows from Green’s
theorem that
(
C 3 −3 ) − 3 = ((− 3 ) −( 3 D −3 ) ) =3 (1 −
D 6 2 − 2 ) From this we see easily that the line integral is maximized if the region D is given by
2
+ 2 ≤ 1, which is the unit disk and hence C is the unit circle positively oriented. In
this case, the maximum value is
2π 3 (1 −
2 + 2 ≤1 2 − 2 ) =3 1 θ
0 (1 −
0 7 2 ) 3π
11
= 6π ( − ) =
24
2 :)...
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This document was uploaded on 02/10/2014.
 Spring '13

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