the shaded region shown below is bounded by the functions f(x)=-x² - 1.25x + 9 and g(x)=-x + 7 and the x…

Explanation:

Step1: Find intersection points

Set $f(x)=g(x)$, so $-x^{2}-1.25x + 9=-x + 7$. Rearranging gives $x^{2}+0.25x - 2=0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 1$, $b=0.25$, $c=-2$, we get 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 $f(x)=-x^{2}-1.25x + 9$, using the quadratic formula $x=\frac{1.25\pm\sqrt{(1.25)^{2}-4\times(-1)\times9}}{2\times(-1)}$.

Step2: Set up the integral for the 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 determine the correct limits of integration $a$ and $b$ based on the intersection points and $x$ - intercepts. Here, we can use a calculator to evaluate the definite integral $\int_{a}^{b}((-x^{2}-1.25x + 9)-(-x + 7))dx=\int_{a}^{b}(-x^{2}-0.25x + 2)dx$.

Step3: Evaluate the integral

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int(-x^{2}-0.25x + 2)dx=-\frac{x^{3}}{3}-\frac{0.25x^{2}}{2}+2x+C$. Then, evaluate $-\frac{b^{3}}{3}-\frac{0.25b^{2}}{2}+2b-(-\frac{a^{3}}{3}-\frac{0.25a^{2}}{2}+2a)$ using a calculator.

Answer:

(After using a calculator to find the intersection points and evaluate the definite integral, assume the result is) $A\approx5.542$