Задача 62292 Вычислить площадь фигуры, ограниченной...
Условие
Решение
x= ρ cos φ
y= ρ sin φ
x^2+y^2= ρ ^2
x^2-4x+y^2=0 ⇒ ρ ^2-4ρ cos φ =0 ⇒ ρ =4cos φ
x^2-8x+y^2=0 ⇒ ρ ^2-8ρ cos φ =0 ⇒ ρ =8cos φ
4cos φ ≤ p ≤ 8cos φ
0 ≤ φ ≤ π/4
S= ∫ ∫_(D) dxdy
D:
x^2-4x+y^2=0 ⇒ ρ ^2-4ρ cos φ =0 ⇒ ρ =4cos φ
x^2-8x+y^2=0 ⇒ ρ ^2-8ρ cos φ =0 ⇒ ρ =8cos φ
4cos φ ≤ p ≤ 8cos φ
0 ≤ φ ≤ π/4
dxdy= ρ d ρ d φ
S= ∫^(π/4)_(0) ∫^(8cos φ _(4cos φ ) ρ d ρ d φ =
=∫^(π/4)_(0) ( ρ ^2/2)|^(8cos φ _(4cos φ ) d φ =
=∫^(π/4)_(0) ( (8cos ^2 φ) /2)-(4cos ^2 φ) /2) ) d φ =
=24∫^(π/4)_(0) ( cos ^2 φ) d φ =
=24∫^(π/4)_(0) (1+ cos 2 φ)/2 d φ =
=12(1+(1/2)sin2 φ )|^(π/4)_(0) =
12*(π/4)+6sin(π/2)-6sin0=[b]3π+6[/b]