Помогите, пожалуйста, найти координаты центра тяжести фигуры, ограниченной линиями
y = x^2 + 4 * x, y = x + 4.

y = x^2 + 4 * x, y = x + 4
Решение.
Найдем точки пересечения графиков этих двух функций:
y = x^2 + 4 * x, y = x + 4
x^2 + 4 * x = x + 4 => x^2 + 3 * x - 4 = 0
D = 25, x1 = -4, x2 = 1
При -4 <= x <= 1 x^2 + 4 * x <= x + 4
Тогда
M = int (-4 1) dx int (x^2 + 4 * x x + 4) dy = int (-4 1) dx (y)_{x^2 + 4 * x}^{x + 4} =
= int (-4 1) dx (x + 4 - x^2 - 4 * x) = int (-4 1) (-x^2 - 3 * x + 4) dx =
= (-1/3 * x^3 - 3 * 1/2 * x^2 + 4 * x)_{-4}^{1} =
= (-1/3 * x^3 - 3/2 * x^2 + 4 * x)_{-4}^{1} =
= (-1/3 * 1^3 - 3/2 * 1^2 + 4 * 1) - (-1/3 * (-4)^3 - 3/2 * (-4)^2 + 4 * (-4)) =
= (-1/3 - 3/2 + 4) - (64/3 - 48/2 - 16) = -1/3 - 3/2 + 4 - 64/3 + 48/2 + 16 =
= -65/3 - 3/2 + 4 + 24 + 16 = -65/3 - 3/2 + 44 = -130/6 - 9/6 + 264/6 = 125/6
x_0 = 1/M * int (-4 1) dx int (x^2 + 4 * x x + 4) x dy =
= 6/125 * int (-4 1) x dx (y)_{x^2 + 4 * x}^{x + 4} =
= 6/125 * int (-4 1) x dx (x + 4 - x^2 - 4 * x) =
= 6/125 * int (-4 1) x * (-x^2 - 3 * x + 4) dx =
= 6/125 * int (-4 1) (-x^3 - 3 * x^2 + 4 * x) dx =
= 6/125 * (-1/4 * x^4 - 3 * 1/3 * x^3 + 4 * 1/2 * x^2)_{-4}^{1} =
= 6/125 * (-1/4 * x^4 - x^3 + 2 * x^2)_{-4}^{1} =
= 6/125 * ((-1/4 * 1^4 - 1^3 + 2 * 1^2) - (-1/4 * (-4)^4 - (-4)^3 + 2 * (-4)^2)) =
= 6/125 * ((-1/4 - 1 + 2) - (-64 + 64 + 32)) = 6/125 * (-1/4 - 1 + 2 + 64 - 64 - 32) =
= 6/125 * (-1/4 + 1 - 32) = 6/125 * (-1/4 - 31) = 6/125 * (-1/4 - 124/4) =
= 6/125 * (-125/4) = -6/4 = -3/2
x_0 = -3/2.
y_0 = 1/M * int (-4 1) dx int (x^2 + 4 * x x + 4) y dy =
= 6/125 * int (-4 1) dx (1/2 * y^2)_{x^2 + 4 * x}^{x + 4} =
= 3/125 * int (-4 1) dx ((x + 4)^2 - (x^2 + 4 * x)^2) =
= 3/125 * int (-4 1) dx ((x^2 + 8 * x + 16) - (x^4 + 8 * x^3 + 16 * x^2)) =
= 3/125 * int (-4 1) (x^2 + 8 * x + 16 - x^4 - 8 * x^3 - 16 * x^2) dx =
= 3/125 * int (-4 1) (-x^4 - 8 * x^3 - 15 * x^2 + 8 * x + 16) dx =
= 3/125 * (-1/5 * x^5 - 8 * 1/4 * x^4 - 15 * 1/3 * x^3 +
+ 8 * 1/2 * x^2 + 16 * x)_{-4}^{1} =
= 3/125 * (-1/5 * x^5 - 2 * x^4 - 5 * x^3 + 4 * x^2 + 16 * x)_{-4}^{1} =
= 3/125 * ((-1/5 * 1^5 - 2 * 1^4 - 5 * 1^3 + 4 * 1^2 + 16 * 1) -
- (-1/5 * (-4)^5 - 2 * (-4)^4 - 5 * (-4)^3 + 4 * (-4)^2 + 16 * (-4))) =
= 3/125 * ((-1/5 - 2 - 5 + 4 + 16) - (1024/5 - 2 * 256 + 5 * 64 + 4 * 16 - 64)) =
= 3/125 * (-1/5 - 2 - 5 + 4 + 16 - 1024/5 + 2 * 256 - 5 * 64 - 4 * 16 + 64) =
= 3/125 * (-1/5 - 7 + 20 - 1024/5 + 512 - 320 - 64 + 64) =
= 3/125 * (-1/5 - 1024/5 + 205) = 3/125 * (-1025/5 + 205) = 3/125 * (-205 + 205) = 0
y_0 = 0
Ответ: x_0 = -3/2, y_0 = 0.