Вычислить площадь фигуры ограниченной линиями y=0,5x^2-2x+3, y=7-x

Y=0,5*x²-2x+3 y=7-x

0,5*x²-2x+3=7-x

0,5*x²-x-4=0 |*2

x²-2x-8=0 D=36 √D=6

x₁=-2 x₂=4 ⇒

₋₂∫⁴ (7-x - (0,5x²-2x+3)) dx=₋₂∫⁴ (7-x-0,5x²+2x-3) dx=₋₂∫⁴ (4+x-0,5x²) dx=

= (1/2) * ₋₂∫⁴ (8+2x-x²) dx = (1/2) * (8x+x²-x³/3) ₋₂|⁴=

= (1/2) (8*4+4²-4³/3 - (8 * (-2) + (-2) ² - (-2) ³/3)) = (1/2) (32+16-64/3 - (-16+4+8/3)) =

= (1/2) (48-64/3 - (-12+8/3)) = (48-64/3+12-8/3) / 2 = (60-72/3) / 2 = (60-24) / 2=36/2=18.

Ответ: S=18 кв. ед.