find the net signed area between the graph of the function f(x)=−x−4 and the x - axis over the interval…

Explanation:

Step1: Recall the definite - integral formula for net - signed area

The net - signed area (A) between the graph of (y = f(x)) and the (x) - axis over the interval ([a,b]) is given by (A=\int_{a}^{b}f(x)dx). Here, (a=-8), (b = 1), and (f(x)=-x - 4). So, (A=\int_{-8}^{1}(-x - 4)dx).

Step2: Use the integral rules

We know that (\int_{-8}^{1}(-x - 4)dx=-\int_{-8}^{1}xdx-\int_{-8}^{1}4dx). The power rule for integration is (\int x^n dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)). For (\int_{-8}^{1}xdx), using the power rule with (n = 1), we have (\int_{-8}^{1}xdx=\left[\frac{x^{2}}{2}\right]{-8}^{1}=\frac{1^{2}}{2}-\frac{(-8)^{2}}{2}=\frac{1}{2}-\frac{64}{2}=-\frac{63}{2}). For (\int{-8}^{1}4dx), since (\int kdx=kx + C) (where (k) is a constant), then (\int_{-8}^{1}4dx=4x\big|_{-8}^{1}=4\times(1-( - 8))=4\times9 = 36).

Step3: Calculate the net - signed area

(A=-\left(-\frac{63}{2}\right)-36=\frac{63}{2}-36=\frac{63 - 72}{2}=-\frac{9}{2}).

Answer:

(-\frac{9}{2})