question find the net signed area between the graph of the function f(x)=|x - 2|-3 and the x - axis over the…

Explanation:

Step1: Rewrite the absolute - value function

When (x - 2\geq0) (i.e., (x\geq2)), (f(x)=(x - 2)-3=x - 5); when (x - 2<0) (i.e., (x<2)), (f(x)=-(x - 2)-3=-x - 1).

Step2: Split the integral based on the break - point

We split the integral (\int_{-4}^{7}(|x - 2|-3)dx) into (\int_{-4}^{2}(-x - 1)dx+\int_{2}^{7}(x - 5)dx).

Step3: Integrate (\int_{-4}^{2}(-x - 1)dx)

Using the power rule (\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (\int_{-4}^{2}(-x - 1)dx=\left[-\frac{x^{2}}{2}-x\right]_{-4}^{2}). [ \begin{align*} &(-\frac{2^{2}}{2}-2)-(-\frac{(-4)^{2}}{2}+4)\ =&(-2 - 2)-(-8 + 4)\ =&-4+4\ =&0 \end{align*} ]

Step4: Integrate (\int_{2}^{7}(x - 5)dx)

Using the power rule, (\int_{2}^{7}(x - 5)dx=\left[\frac{x^{2}}{2}-5x\right]_{2}^{7}). [ \begin{align*} &(\frac{7^{2}}{2}-5\times7)-(\frac{2^{2}}{2}-5\times2)\ =&(\frac{49}{2}-35)-(\ 2 - 10)\ =&\frac{49}{2}-35 - 2 + 10\ =&\frac{49}{2}-27\ =&\frac{49-54}{2}\ =&-\frac{5}{2} \end{align*} ]

Step5: Sum the two integral results

The net - signed area (A = 0-\frac{5}{2}=-\frac{5}{2}).

Answer:

(-\frac{5}{2})