question find the net signed area between the graph of the function f(x)=-5x/4 - 6 and the x - axis over the…

Explanation:

Step1: Recall the net - signed area formula

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, $a =-\frac{34}{5}$, $b=\frac{11}{5}$, and $f(x)=-\frac{5x}{4}-6$.

Step2: Calculate the integral

We know that $\int(-\frac{5x}{4}-6)dx=-\frac{5}{4}\int xdx-6\int dx$. Using the power rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int xdx=\frac{x^{2}}{2}$ and $\int dx=x$. So, $\int(-\frac{5x}{4}-6)dx=-\frac{5}{8}x^{2}-6x + C$.

Step3: Evaluate the definite integral

[ \begin{align*} \int_{-\frac{34}{5}}^{\frac{11}{5}}(-\frac{5x}{4}-6)dx&=\left(-\frac{5}{8}x^{2}-6x\right)\Big|_{-\frac{34}{5}}^{\frac{11}{5}}\ &=\left(-\frac{5}{8}\times(\frac{11}{5})^{2}-6\times\frac{11}{5}\right)-\left(-\frac{5}{8}\times(-\frac{34}{5})^{2}-6\times(-\frac{34}{5})\right)\ &=\left(-\frac{5}{8}\times\frac{121}{25}-\frac{66}{5}\right)-\left(-\frac{5}{8}\times\frac{1156}{25}+\frac{204}{5}\right)\ &=\left(-\frac{121}{40}-\frac{66}{5}\right)-\left(-\frac{1156}{40}+\frac{204}{5}\right)\ &=\left(-\frac{121 + 528}{40}\right)-\left(-\frac{1156+1632}{40}\right)\ &=\frac{- 649}{40}-\frac{-2788}{40}\ &=\frac{-649 + 2788}{40}\ &=\frac{2139}{40} \end{align*} ]

Answer:

$\frac{2139}{40}$