Then 2 2 2 notice that 0 works and hence

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Unformatted text preview: 2 2 + ( 2 + Notice that ( ) = 0 works and hence ( Newton-Leibniz’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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