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. ∫₂¹² f(x)dx submit your answer as an exact value.

question the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫₂¹² f(x)dx submit your answer as an exact value.

Answer

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})