Graph the two curves with your calculator.
Notice that from \(1\lt x\lt 2\), the polynomial is above the line. From \(2\lt x\lt 5\), the line is above the polynomial.
The area can be found using:
$$ \small \int\limits_1^2 (x^3-8x^2+18x-5)-(x+5) dx + \int\limits_2^5 (x+5)-(x^3-8x^2+18x-5) dx $$
$$ = \int\limits_1^2 x^3-8x^2+17x-10 dx + \int\limits_2^5 -x^3+8x^2-17x+10 dx $$
$$ \approx 0.583+11.25 $$
$$ = \boxed{11.833} $$
Since you graphed both functions, your graphing calculator has access to them. To avoid typing in a lot of variables and numbers, you could do this (depending on which functions you entered for \(Y_1\) and \(Y_2\)):
$$ \int\limits_1^2 Y_1-Y_2 dx + \int\limits_2^5 Y_2-Y_1 dx $$
What is the area enclosed by the curves \(y=x^3-8x^2+18x-5\) and \(y=x+5\) ?