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

the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫₁⁹ f(x) dx submit your answer as an exact value. provide your answer below: ∫₁⁹ f(x)=□

the piece - wise function f(x) is graphed below. use geometric formulas to evaluate the following definite integral. ∫₁⁹ f(x) dx submit your answer as an exact value. provide your answer below: ∫₁⁹ f(x)=□

Answer

Explanation:

Step1: Divide the region

The region under the curve from (x = 1) to (x=9) can be divided into geometric - shapes. We have two triangles.

Step2: Calculate the area of the first triangle

The first triangle has a base (b_1=4) and height (h_1 = 4). The area of a triangle is (A=\frac{1}{2}bh). So, (A_1=\frac{1}{2}\times4\times4 = 8).

Step3: Calculate the area of the second triangle

The second triangle has a base (b_2 = 4) and height (h_2=2). So, (A_2=\frac{1}{2}\times4\times2=4).

Step4: Calculate the definite - integral

The definite integral (\int_{1}^{9}f(x)dx) is the sum of the areas of the two triangles. Since the part of the graph below the (x) - axis has a negative contribution and the part above has a positive contribution, (\int_{1}^{9}f(x)dx=-A_1 + A_2). [ \begin{align*} \int_{1}^{9}f(x)dx&=- 8+4\ &=-4 \end{align*} ]

Answer:

(-4)