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

# Explanation: ## Step1: Divide the region The region under the curve from \(x = 1\) to \(x=11\) can be divided into a triangle from \(x = 1\) to \(x = 3\), a triangle from \(x=3\) to \(x = 5\), a rectangle from \(x = 5\) to \(x=9\) and a triangle from \(x = 9\) to \(x = 11\). ## Step2: Calculate area of first - triangle The base of the first triangle from \(x = 1\) to \(x = 3\) is \(b_1=3 - 1=2\), and the height \(h_1=- 1\). The area of a triangle is \(A=\frac{1}{2}bh\), so \(A_1=\frac{1}{2}\times2\times(-1)=-1\). ## Step3: Calculate area of second - triangle The base of the second triangle from \(x = 3\) to \(x = 5\) is \(b_2=5 - 3 = 2\), and the height \(h_2 = 1\). So \(A_2=\frac{1}{2}\times2\times1 = 1\). ## Step4: Calculate area of rectangle The base of the rectangle from \(x = 5\) to \(x=9\) is \(b_3=9 - 5=4\), and the height \(h_3 = 1\). The area of a rectangle is \(A = bh\), so \(A_3=4\times1=4\). ## Step5: Calculate area of third - triangle The base of the third triangle from \(x = 9\) to \(x = 11\) is \(b_4=11 - 9=2\), and the height \(h_4 = 1\). So \(A_4=\frac{1}{2}\times2\times1=1\). ## Step6: Sum up the areas \(\int_{1}^{11}f(x)dx=A_1 + A_2+A_3+A_4=-1 + 1+4 + 1\). # Answer: 4