the shaded region shown below is bounded by the functions f(x)=-3x² + 9 and g(x)=-1.5x + 7 and the x and y…

Explanation:

Step1: Find intersection point

Set $f(x)=g(x)$, so $-3x^{2}+9=-1.5x + 7$. Rearrange to $3x^{2}-1.5x - 2=0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 3$, $b=-1.5$, $c=-2$, we get the positive - valued intersection point $x$ (since we are in the first - quadrant).

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 $x_0$ is $A=\int_{0}^{x_0}((-3x^{2}+9)-(-1.5x + 7))dx=\int_{0}^{x_0}(-3x^{2}+1.5x + 2)dx$.

Step3: Evaluate integral

Using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int(-3x^{2}+1.5x + 2)dx=-x^{3}+0.75x^{2}+2x+C$. Evaluate the definite integral $A=\left[-x^{3}+0.75x^{2}+2x\right]_{0}^{x_0}$. Using a calculator to solve the quadratic equation $3x^{2}-1.5x - 2=0$ for the positive root $x_0=\frac{1.5+\sqrt{(-1.5)^{2}-4\times3\times(-2)}}{2\times3}\approx1.089$. Then $A=- (1.089)^{3}+0.75\times(1.089)^{2}+2\times1.089\approx - 1.288+0.887 + 2.178=1.777$.

Answer:

$1.777$