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ρ 3 cos θ = aρ 2 (cos 2 θ − sin 2 θ)<br />
<br />
cos 2θ<br />
ρ = a<br />
cos θ<br />
θ ∈ [−π/4, π/4] ∪ [π/2, 3/4π] ∪ [5/4π, 3/2π] x(θ) = ρ cos θ = a cos 2θ <br />
<br />
lim ρ = +∞ x = −a ρ(θ) = 0 θ =<br />
θ→±π/2∓ ±π/4 −π/4 < θ < π/4 0 < x < a |y| ≤<br />
x (a − x)/(a + x) C C <br />
<br />
<br />
C<br />
<br />
C<br />
dx dy =<br />
x dx dy =<br />
π/4 a cos(2θ)/ cos θ<br />
= a2<br />
2<br />
= a2<br />
2<br />
= a2<br />
2<br />
= a2<br />
2<br />
−π/4<br />
0<br />
π/4<br />
<br />
<br />
2<br />
2<br />
4 cos<br />
−π/4<br />
2 θ + 1<br />
π/4<br />
−π/4<br />
π/4<br />
−π/4<br />
(4 − π)<br />
π/4 a cos(2θ)/ cos θ<br />
= a3<br />
3<br />
= a3<br />
3<br />
= a3<br />
3<br />
= a3<br />
3<br />
= a3<br />
3<br />
−π/4 0<br />
π/4<br />
s ds dθ = a2<br />
2<br />
cos2 − 4 dθ<br />
θ<br />
π/4<br />
−π/4<br />
<br />
(cos 2θ + 1) dθ + [tan θ] π/4<br />
cos 2θ dθ + 2 − π<br />
<br />
−π/4<br />
π/4<br />
cos 2 (2θ)<br />
cos 2 θ dθ<br />
− 2π<br />
s 2 cos θ ds dθ = a3<br />
3 −π/4<br />
<br />
2 cos θ −<br />
−π/4<br />
1<br />
3 cos θ dθ<br />
cos θ<br />
π/4 <br />
8 cos<br />
−π/4<br />
4 θ − 12 cos 2 θ − 1<br />
cos2 <br />
+ 6 dθ<br />
θ<br />
<br />
π/4<br />
2(cos 2θ + 1)<br />
−π/4<br />
2 dθ − 3(2 + π) − 2 + 3π<br />
π/4<br />
<br />
cos 3 (2θ)<br />
cos 2 θ dθ<br />
2(cos<br />
−π/4<br />
2 2θ + 2 cos 2θ + 1) dθ − 3(2 + π) − 2 + 3π<br />
π/4<br />
(cos 4θ + 1 + 4 cos 2θ) dθ + π − 8<br />
−π/4<br />
= a3<br />
π/4<br />
(cos 4θ) dθ +<br />
3 −π/4<br />
3<br />
<br />
π − 4<br />
2<br />
= a3<br />
<br />
3<br />
π − 4 =<br />
3 2 a3<br />
(3π − 8)<br />
6<br />
<br />
<br />
x dx dy<br />
C<br />
xB = =<br />
dx dy<br />
a 3π − 8<br />
3 4 − π , yB = 0.<br />
C
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