question find the net signed area between 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: $\frac{2139}{40}$