current objective find the average value of a function over an interval question find the average value of…

Explanation:

Step1: Recall average - value formula

The average value of a function $y = g(x)$ over the interval $[a,b]$ is given by $\frac{1}{b - a}\int_{a}^{b}g(x)dx$. Here, $a=-7$, $b = 1$, and $g(x)=-6x + 3$. So we need to calculate $\frac{1}{1-(-7)}\int_{-7}^{1}(-6x + 3)dx=\frac{1}{8}\int_{-7}^{1}(-6x + 3)dx$.

Step2: Integrate the function

We know that $\int(-6x + 3)dx=-6\int xdx+3\int dx$. Using the power - rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int(-6x + 3)dx=-6\times\frac{x^{2}}{2}+3x=-3x^{2}+3x$.

Step3: Evaluate the definite integral

$\int_{-7}^{1}(-6x + 3)dx=\left[-3x^{2}+3x\right]{-7}^{1}$. First, substitute $x = 1$: $-3(1)^{2}+3(1)=-3 + 3=0$. Then substitute $x=-7$: $-3(-7)^{2}+3(-7)=-3\times49-21=-147-21=-168$. So, $\int{-7}^{1}(-6x + 3)dx=0-(-168)=168$.

Step4: Calculate the average value

The average value is $\frac{1}{8}\times168 = 21$.

Answer:

$21$