question 9. 1 point given the function f(x)=-x - 2, find the net signed area between f(x) and the x - axis…

Explanation:

Step1: Recall the integral formula for net - signed area

The net - signed area (A) between a function (y = f(x)) and the (x) - axis over the interval ([a,b]) is given by (A=\int_{a}^{b}f(x)dx). Here, (a=-5), (b = 3) and (f(x)=-x - 2). So, (A=\int_{-5}^{3}(-x - 2)dx).

Step2: Use the sum rule of integration

(\int_{-5}^{3}(-x - 2)dx=\int_{-5}^{3}-xdx+\int_{-5}^{3}- 2dx).

Step3: Integrate each term separately

For (\int_{-5}^{3}-xdx), using the power rule (\int x^n dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have (\int - xdx=-\frac{x^{2}}{2}+C). Then (\int_{-5}^{3}-xdx=-\left[\frac{x^{2}}{2}\right]{-5}^{3}=-\left(\frac{3^{2}}{2}-\frac{(-5)^{2}}{2}\right)=-\left(\frac{9}{2}-\frac{25}{2}\right)=-\frac{9 - 25}{2}=8). For (\int{-5}^{3}-2dx), since (\int kdx=kx + C) ((k) is a constant), we have (\int_{-5}^{3}-2dx=-2[x]_{-5}^{3}=-2(3-(-5))=-2\times8=-16).

Step4: Combine the results

(A=\int_{-5}^{3}-xdx+\int_{-5}^{3}-2dx=8+( - 16)=-8).

Answer:

(-8)