the regions a, b, c, d, and e in the figure below are bounded by the graph of the function f and the x…

Explanation:

Step1: Recall integral - area relationship

The definite integral $\int_{a}^{b}f(x)dx$ is the net - signed area between the graph of $y = f(x)$ and the $x$ - axis. Areas above the $x$ - axis contribute positively and areas below the $x$ - axis contribute negatively.

Step2: Identify positive and negative areas

Regions A, C, and E are above the $x$ - axis, so their areas contribute positively. Regions B and D are below the $x$ - axis, so their areas contribute negatively. The area of region A is $A_A=4$, the area of region B is $A_B = 5$, the area of region C is $A_C=6$, the area of region D is $A_D = 6$, and the area of region E is $A_E=5$.

Step3: Calculate the definite integral

$\int_{-2}^{7}f(x)dx=A_A - A_B+A_C - A_D+A_E$. Substitute the values: $\int_{-2}^{7}f(x)dx=4 - 5+6 - 6+5$. First, $4 - 5=-1$, then $-1 + 6 = 5$, then $5-6=-1$, and finally $-1 + 5 = 4$.

Answer:

$4$