question the piece - wise function f(x) is gr...

# Explanation: ## Step1: Divide the region The region under the curve from $x = 2$ to $x=12$ can be divided into 3 geometric - shapes: a triangle from $x = 2$ to $x = 6$, a rectangle from $x = 6$ to $x = 10$, and a triangle from $x = 10$ to $x = 12$. ## Step2: Calculate area of first triangle The base of the first triangle from $x = 2$ to $x = 6$ is $b_1=6 - 2=4$, and the height $h_1=- 3$. The area of a triangle is $A=\frac{1}{2}bh$. So, $A_1=\frac{1}{2}\times4\times(-3)=-6$. ## Step3: Calculate area of rectangle The base of the rectangle from $x = 6$ to $x = 10$ is $b_2=10 - 6 = 4$, and the height $h_2 = 3$. The area of a rectangle is $A = bh$. So, $A_2=4\times3 = 12$. ## Step4: Calculate area of second triangle The base of the second triangle from $x = 10$ to $x = 12$ is $b_3=12 - 10=2$, and the height $h_3 = 3$. The area of a triangle is $A=\frac{1}{2}bh$. So, $A_3=\frac{1}{2}\times2\times3 = 3$. ## Step5: Calculate the definite - integral The definite integral $\int_{2}^{12}f(x)dx$ is the sum of the areas of these geometric shapes. $\int_{2}^{12}f(x)dx=A_1 + A_2+A_3=-6 + 12+3$. # Answer: 9