int (4 + x^2 + y^2)^(1/2) dx dy, 1 <= x^2 + y^2 <= 4
Перейдем к полярным координатам:
x = r * cos fi, y = r * sin fi
x^2 + y^2 = r^2 * cos^2 fi + r^2 * sin^2 fi = r^2
Тогда 4 + x^2 + y^2 = 4 + r^2 и dx dy = r dr dfi.
Найдем пределы интегрирования по r и fi: 1 <= x^2 + y^2 <= 4 (кольцо)
1 <= r^2 <= 4 => 1 <= r <= 2 и 0 <= fi <= 2 * pi.
Тогда
int (4 + x^2 + y^2)^(1/2) dx dy = int (0 2pi) dfi int (1 2) (4 + r^2)^(1/2) * r dr =
= 2 * pi * int (1 2) (4 + r^2)^(1/2) d(1/2 * r^2) =
= 2 * pi * 1/2 * int (1 2) (4 + r^2)^(1/2) d(r^2) =
= pi * int (1 2) (4 + r^2)^(1/2) d(4 + r^2) = | 4 + r^2 = t | =
= pi * int (5 8) t^(1/2) dt = pi * (1/(1/2 + 1) * t^(1/2 + 1))_{5}^{8} =
= pi * (2/3 * t^(3/2))_{5}^{8} = pi * (2/3 * 8^(3/2) - 2/3 * 5^(3/2)) =
= 2/3 * pi * (8 * 8^(1/2) - 5 * 5^(1/2))