the piecewise function f(x) is graphed below. use geometric formulas to evaluate the following definite…

Explanation:

Step1: Divide the region

The definite - integral $\int_{1}^{9}f(x)dx$ can be divided into three geometric shapes: two triangles and one trapezoid.

Step2: Calculate the area of the first triangle

The first triangle has a base from $x = 1$ to $x = 4$ and height $h = 2$. The area of a triangle is $A_1=\frac{1}{2}\times base\times height$. Here, $base = 4 - 1=3$ and $height = 2$, so $A_1=\frac{1}{2}\times3\times2 = 3$.

Step3: Calculate the area of the trapezoid

The trapezoid has bases $b_1 = 2$ and $b_2=0$ and height $h = 1$ (from $x = 4$ to $x = 5$). The area of a trapezoid is $A_2=\frac{(b_1 + b_2)h}{2}$. Here, $A_2=\frac{(2 + 0)\times1}{2}=1$.

Step4: Calculate the area of the second triangle

The second triangle has a base from $x = 5$ to $x = 9$ and height $h = 4$. The base length is $9 - 5 = 4$, and the area of the triangle is $A_3=\frac{1}{2}\times4\times4=8$. But since the part of the function below the $x -$axis, its value for the integral is negative, so $A_3=- 8$.

Step5: Sum up the areas

$\int_{1}^{9}f(x)dx=A_1+A_2+A_3$. Substitute the values of $A_1 = 3$, $A_2 = 1$, and $A_3=-8$ into the formula: $\int_{1}^{9}f(x)dx=3 + 1-8=-4$.

Answer:

$-4$