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

Explanation:

Step1: Divide the region

The region under the curve from $x = 2$ to $x=12$ can be divided into three geometric - shapes: a triangle from $x = 2$ to $x = 5$, a triangle from $x = 5$ to $x = 6$, and a trapezoid from $x = 6$ to $x = 12$.

Step2: Calculate the area of the first triangle

The base of the first triangle from $x = 2$ to $x = 5$ is $b_1=5 - 2=3$, and the height $h_1 = 3$. The area of a triangle is $A=\frac{1}{2}bh$. So, $A_1=\frac{1}{2}\times3\times3=\frac{9}{2}$.

Step3: Calculate the area of the second triangle

The base of the second triangle from $x = 5$ to $x = 6$ is $b_2=6 - 5 = 1$, and the height $h_2=3$. So, $A_2=\frac{1}{2}\times1\times3=\frac{3}{2}$.

Step4: Calculate the area of the trapezoid

The trapezoid from $x = 6$ to $x = 12$ has bases $b_1 = 0$ and $b_2=3$, and height $h=6$. The area of a trapezoid is $A=\frac{(b_1 + b_2)h}{2}$. So, $A_3=\frac{(0 + 3)\times6}{2}=9$. Also, since the trapezoid is below the $x$-axis, its contribution to the integral is negative, $A_3=-9$.

Step5: Sum up the areas

The definite - integral $\int_{2}^{12}f(x)dx=A_1+A_2+A_3$. Substitute the values: $\int_{2}^{12}f(x)dx=\frac{9}{2}+\frac{3}{2}-9$. First, add the first two terms: $\frac{9 + 3}{2}-9=\frac{12}{2}-9=6 - 9=-3$.

Answer:

$-3$