(a) what is the total area between the graph of f(x) in the figure and the x - axis, between x = 0 and x =…

Explanation:

Step1: Define total - area concept

The total area between the graph of $y = f(x)$ and the $x$-axis is the sum of the absolute - value of the areas above and below the $x$-axis.

Step2: Calculate total area

The area above the $x$-axis is 7 and the area below the $x$-axis is 6. The total area $A_{total}=|7| + |6|=7 + 6=13$.

Step3: Define definite - integral concept

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 are positive and areas below the $x$-axis are negative.

Step4: Calculate definite integral

$\int_{0}^{5}f(x)dx=7-6 = 1$.

Answer:

(a) 13 (b) 1