the piece - wise function f(x) is graphed bel...
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:
\(-4\)