if the average value of the function f over the closed interval 2, 4 is 3 and if f(x) ≥ 0 for all x in 2, 4…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ over the interval $[a,b]$ is given by $\bar{f}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a = 2$, $b = 4$ and $\bar{f}=3$.

Step2: Find the integral value

We know that $3=\frac{1}{4 - 2}\int_{2}^{4}f(x)dx$. Then $\int_{2}^{4}f(x)dx=3\times(4 - 2)$.

Step3: Calculate the result

$\int_{2}^{4}f(x)dx = 6$. The area of the region enclosed by the graph of $y = f(x)$, the lines $x = 2$, $x = 4$ and the $x$-axis is given by $\int_{2}^{4}f(x)dx$.

Answer:

B. 6