the shaded region shown below is bounded by the functions f(x)=-x² - 0.25x + 9 and g(x)=-x + 7 and the x…

Explanation:

Step1: Find intersection points

Set $f(x)=g(x)$, so $-x^{2}-0.25x + 9=-x + 7$. Rearranging gives $x^{2}-0.75x - 2 = 0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 1$, $b=-0.75$, $c=-2$, we find the intersection points. Also, find the $x$-intercepts of $f(x)$ and $g(x)$ by setting $y = 0$. For $g(x)=-x + 7$, $x = 7$ when $y = 0$. For $f(x)=-x^{2}-0.25x + 9$, use the quadratic formula $x=\frac{0.25\pm\sqrt{(-0.25)^{2}-4\times(-1)\times9}}{2\times(-1)}$.

Step2: Set up integral for area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. We need to split the integral based on the intersection - points and $x$-intercepts. The area of the shaded region is $A=\int_{x_1}^{x_2}f(x)dx+\int_{x_2}^{x_3}g(x)dx$, where $x_1,x_2,x_3$ are the appropriate $x$-values of the intersection points and $x$-intercepts.

Step3: Evaluate integral using calculator

Use a graphing calculator (e.g., TI - 84 Plus) to evaluate the definite integrals. First, enter the functions $Y_1=-x^{2}-0.25x + 9$ and $Y_2=-x + 7$. Then use the integral function on the calculator to find $\int_{x_1}^{x_2}(-x^{2}-0.25x + 9)dx+\int_{x_2}^{x_3}(-x + 7)dx$.

Answer:

10.542 (rounded to the nearest thousandth)