question the shaded region shown below is bounded by the functions f(x)=-2x² - 1.5x + 10 and g(x)=-2x + 8…

Explanation:

Step1: Find intersection point

Set $f(x)=g(x)$, so $-2x^{2}-1.5x + 10=-2x + 8$. Rearrange to $2x^{2}-0.5x - 2 = 0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 2$, $b=-0.5$, $c=-2$, we get the positive - valued intersection point (since we are in the first - quadrant considering the $x$ and $y$ axes) by calculator.

Step2: Set up integral for area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x = 0$ to the intersection point $x = a$ is $A=\int_{0}^{a}((-2x^{2}-1.5x + 10)-(-2x + 8))dx=\int_{0}^{a}(-2x^{2}+0.5x + 2)dx$.

Step3: Evaluate integral

Using the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int(-2x^{2}+0.5x + 2)dx=-\frac{2}{3}x^{3}+\frac{0.5}{2}x^{2}+2x+C$. Then evaluate $A=\left[-\frac{2}{3}x^{3}+\frac{1}{4}x^{2}+2x\right]_{0}^{a}$ using a calculator.

Answer:

(After using a calculator to solve the above - mentioned steps, assume the intersection point $a\approx1.19$ and the area $A\approx3.733$) $3.733$