question find the net signed area between the...
question find the net signed area between the graph of the function f(x)=|x - 2|-3 and the x - axis over the interval -4,7. submit your answer as an exact value. sorry, thats incorrect. try again? net signed area =
Answer
# 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:
\(-\frac{5}{2}\)