question find the net signed area between the...

# Explanation: ## Step1: Recall the formula for net - signed area The net - signed area \(A\) between the graph of \(y = f(x)\) and the \(x\) - axis over the interval \([a,b]\) is given by \(A=\int_{a}^{b}f(x)dx\), where \(a =-\frac{34}{5}\), \(b=\frac{11}{5}\), and \(f(x)=-\frac{5x}{4}-6\). ## Step2: Calculate the integral We know that \(\int\left(-\frac{5x}{4}-6\right)dx=-\frac{5}{4}\int xdx-6\int dx\). Using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n\neq - 1\)), we have \(-\frac{5}{4}\times\frac{x^{2}}{2}-6x+C=-\frac{5x^{2}}{8}-6x+C\). ## Step3: Evaluate the definite integral \(\int_{-\frac{34}{5}}^{\frac{11}{5}}\left(-\frac{5x}{4}-6\right)dx=\left[-\frac{5x^{2}}{8}-6x\right]_{-\frac{34}{5}}^{\frac{11}{5}}\) First, substitute \(x = \frac{11}{5}\): \(-\frac{5}{8}\times\left(\frac{11}{5}\right)^{2}-6\times\frac{11}{5}=-\frac{5}{8}\times\frac{121}{25}-\frac{66}{5}=-\frac{121}{40}-\frac{66}{5}\) \(=-\frac{121}{40}-\frac{528}{40}=-\frac{121 + 528}{40}=-\frac{649}{40}\) Then substitute \(x=-\frac{34}{5}\): \(-\frac{5}{8}\times\left(-\frac{34}{5}\right)^{2}-6\times\left(-\frac{34}{5}\right)=-\frac{5}{8}\times\frac{1156}{25}+\frac{204}{5}\) \(=-\frac{1156}{40}+\frac{204}{5}=-\frac{1156}{40}+\frac{1632}{40}=\frac{-1156 + 1632}{40}=\frac{476}{40}\) Now, \(\left(-\frac{649}{40}\right)-\frac{476}{40}=\frac{-649 - 476}{40}=-\frac{1125}{40}=-\frac{225}{8}\) # Answer: \(-\frac{225}{8}\)