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 three geometric - shapes: two triangles and one rectangle. ## Step2: Analyze the first triangle The first triangle has a base \(b_1=4\) (from \(x = 2\) to \(x = 6\)) and height \(h_1 = 3\). The area of a triangle is \(A=\frac{1}{2}bh\). So, \(A_1=\frac{1}{2}\times4\times3=6\). ## Step3: Analyze the rectangle The rectangle has a base \(b_2 = 3\) (from \(x = 6\) to \(x = 9\)) and height \(h_2=- 5\). The area of a rectangle is \(A = bh\). So, \(A_2=3\times(-5)=- 15\). ## Step4: Analyze the second triangle The second triangle has a base \(b_3 = 3\) (from \(x = 9\) to \(x = 12\)) and height \(h_3 = 5\). The area of a triangle is \(A=\frac{1}{2}bh\). So, \(A_3=\frac{1}{2}\times3\times5=\frac{15}{2}\). ## Step5: Calculate the definite - integral The definite integral \(\int_{2}^{12}f(x)dx=A_1 + A_2+A_3\). \[ \begin{align*} \int_{2}^{12}f(x)dx&=6-15+\frac{15}{2}\\ &=\frac{12 - 30 + 15}{2}\\ &=-\frac{3}{2} \end{align*} \] # Answer: \(-\frac{3}{2}\)