question the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following…

Explanation:

Step1: Divide the region

The region under the curve of $y = f(x)$ from $x = 1$ to $x=9$ can be divided into three geometric - shapes: two triangles and one trapezoid.

Step2: Analyze the first triangle

The first triangle has a base from $x = 1$ to $x = 4$. The base length $b_1=4 - 1=3$, and the height $h_1 = 2$. The area of a triangle is $A=\frac{1}{2}bh$. So, $A_1=\frac{1}{2}\times3\times2 = 3$.

Step3: Analyze the trapezoid

The trapezoid has bases $b_1 = 2$ and $b_2=4$, and height $h = 1$ (from $x = 4$ to $x = 5$). The area of a trapezoid is $A=\frac{(b_1 + b_2)h}{2}$. So, $A_2=\frac{(2 + 4)\times1}{2}=3$.

Step4: Analyze the second triangle

The second triangle has a base from $x = 5$ to $x = 9$. The base length $b_3=9 - 5 = 4$, and the height $h_3=- 4$ (negative because it is below the $x$-axis). The area of a triangle is $A=\frac{1}{2}bh$. So, $A_3=\frac{1}{2}\times4\times(-4)=-8$.

Step5: Calculate the definite - integral

The definite integral $\int_{1}^{9}f(x)dx$ is the sum of the areas of these geometric shapes. $\int_{1}^{9}f(x)dx=A_1+A_2+A_3$. [ \begin{align*} \int_{1}^{9}f(x)dx&=3 + 3-8\ &=-2 \end{align*} ]

Answer:

$-2$