[tex]\displaystyle\\OTBET:\ \frac{1000}{3} .[/tex]

Объяснение:

[tex]\displaystyle\\y=x^2-6x-24\ \ \ \ y=6x+8-x^2\ \ \ \ S=?\\\\x^2-6x-24=6x+8-x^2\\\\2x^2-12x-32=0\ |:2\\\\x^2-6x-16=0\\\\x^2-8x+2x-16=0\\\\x*(x-8)+2*(x-8)=0\\\\(x-8)*(x+2)=0\\\\x-8=0\\\\x_1=8.\\\\x+2=0\\\\x_2=-2 \ \ \ \ \ \ \Rightarrow\\[/tex]

[tex]\displaystyle\\S=\int\limits^8_{-2} {(6x+8-x^2-(x^2-6x-24))} \, dx =\int\limits^8_{-2} {(-2x^2+12x+32)} \, dx=\\\\=-2*\int\limits^8_{-2} {(x^2-6x-16)} \, dx =2*(\frac{x^3}{3} -3x^2-16x)\ |_{-2}^8=\\\\=-2*(\frac{8^3}{3} -3*8^2-16*8-(\frac{(-2)^3}{3}-3*(-2)^2-16*(-2))=\\\\=-2*(\frac{512}{3} -192-128+\frac{8}{3} +12-32)=-2*(\frac{520}{3} -340)=\\\\=-2*(-\frac{520-1020}{3} )= -2*(-\frac{500}{3})=\frac{1000}{3} .[/tex]