question find the net signed area between the...
question find the net signed area between the graph of the function f(x)=|x + 3|-3 and the x - axis over the interval -9,2. submit your answer as an exact value. provide your answer below:
Answer
# Explanation:
## Step1: Rewrite absolute - value function
The absolute - value function \(y = |x + 3|-3\) can be rewritten as a piece - wise function. When \(x+3\geq0\) (i.e., \(x\geq - 3\)), \(y=(x + 3)-3=x\); when \(x+3<0\) (i.e., \(x<-3\)), \(y=-(x + 3)-3=-x - 6\).
## Step2: Split the integral based on the break - point
We split the integral \(\int_{-9}^{2}(|x + 3|-3)dx\) into two integrals based on \(x=-3\). So \(\int_{-9}^{2}(|x + 3|-3)dx=\int_{-9}^{-3}(-x - 6)dx+\int_{-3}^{2}x dx\).
## Step3: Integrate the first integral
Using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)\), for \(\int_{-9}^{-3}(-x - 6)dx=\left[-\frac{x^{2}}{2}-6x\right]_{-9}^{-3}\).
\[
\begin{align*}
&(-\frac{(-3)^{2}}{2}-6\times(-3))-(-\frac{(-9)^{2}}{2}-6\times(-9))\\
=&(-\frac{9}{2}+18)-(-\frac{81}{2}+54)\\
=&-\frac{9}{2}+18+\frac{81}{2}-54\\
=&\frac{-9 + 81}{2}+18-54\\
=&\frac{72}{2}+18-54\\
=&36 + 18-54\\
=&0
\end{align*}
\]
## Step4: Integrate the second integral
For \(\int_{-3}^{2}x dx=\left[\frac{x^{2}}{2}\right]_{-3}^{2}=\frac{2^{2}}{2}-\frac{(-3)^{2}}{2}=\frac{4}{2}-\frac{9}{2}=-\frac{5}{2}\).
## Step5: Sum the results of the two integrals
\(\int_{-9}^{2}(|x + 3|-3)dx=0-\frac{5}{2}=-\frac{5}{2}\).
# Answer:
\(-\frac{5}{2}\)