the graph of function f is given along with the area of each region the graph forms with the x - axis. ∫₀⁷…

Explanation:

Step1: Recall definite - integral property

The definite integral $\int_{a}^{b}f(x)dx$ is the net - signed area between the curve $y = f(x)$ and the $x$ - axis from $x=a$ to $x = b$. Areas above the $x$ - axis are positive and areas below the $x$ - axis are negative.

Step2: Analyze the regions from $x = 0$ to $x = 7$

From $x=0$ to $x = 4$, the area between the curve and the $x$ - axis is below the $x$ - axis and has an area of $7$. From $x = 4$ to $x = 7$, the area between the curve and the $x$ - axis is above the $x$ - axis and has an area of $2$.

Step3: Calculate the definite integral

$\int_{0}^{7}f(x)dx=-7 + 2$.

Answer:

$-5$