Unformatted text preview: 2 2 + ( 2 + Notice that (
) = 0 works and hence (
NewtonLeibniz’s formula, the integral is
(
E 2 2 2) −( 2 )= + 1 1 1) = + 2 2 )= 2 + 2 + 2 ) is the potential. By − :) (a) Let C be a closed curve. Prove that the area inside C equals
1
−
=−
=
2C
C
C
(b) Apply this formula to ﬁnd the area enclosed by the astroid S +( 2/ 3 + 2/ 3 = 2/ 3 . (a) Let A be the area enclosed by the curve C . By Green’s Theorem,
1
=−
=
−
=
2C
C
C
A
(b) The astroid can be parameterized as
Thus the area enclosed by it is = cos3 and = sin3 for 0 ≤ 2π cos3 =
C ( sin3 ) 0
2π 2π cos3 · 3 sin2 cos = =3 2
0 0
2π =3 2
0
2π =3 (sin cos )2 cos2 2
0 2 sin 2 (1 + cos 2 )
8
(1 − cos 4 )(1 + cos 2 )
16
2 = 3π
8 2 ≤ 2π . Note in the last integra...
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This document was uploaded on 02/10/2014.
 Spring '13

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