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

question the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫₁¹¹f(x)dx submit your answer as an exact value.

Answer

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).