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

the regions a, b, c, and d in the figure below are bounded by the graph of the function f and the x - axis. the area of region a is 13, the area of region b is 12, the area of region c is 14, and the area of region d is 4. what is the average value of f on the interval -4, 7 in simplest form?

the regions a, b, c, and d in the figure below are bounded by the graph of the function f and the x - axis. the area of region a is 13, the area of region b is 12, the area of region c is 14, and the area of region d is 4. what is the average value of f on the interval -4, 7 in simplest form?

Answer

Explanation:

Step1: Recall average - value formula

The average value of a function $y = f(x)$ on the interval $[a,b]$ is given by $\bar{f}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$. Here, $a=-4$, $b = 7$, so $b - a=7-(-4)=11$.

Step2: Calculate the definite - integral

The definite integral $\int_{-4}^{7}f(x)dx$ 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. So, $\int_{-4}^{7}f(x)dx=-13 + 12+14-4$.

Step3: Simplify the integral value

$-13 + 12+14-4=(-13-4)+(12 + 14)=-17 + 26 = 9$.

Step4: Calculate the average value

Using the average - value formula $\bar{f}=\frac{1}{b - a}\int_{a}^{b}f(x)dx$, substitute $b - a = 11$ and $\int_{-4}^{7}f(x)dx = 9$. We get $\bar{f}=\frac{9}{11}$.

Answer:

$\frac{9}{11}$