find the total area between the graph of the ...

# Explanation: ## Step1: Analyze the absolute - value function Rewrite \(y = |x - 3|-2\) as a piece - wise function. When \(x\geq3\), \(y=(x - 3)-2=x - 5\); when \(x<3\), \(y=-(x - 3)-2=-x + 1\). ## Step2: Find the x - intercepts For \(y=-x + 1\), set \(y = 0\), then \(0=-x + 1\), \(x = 1\). For \(y=x - 5\), set \(y = 0\), then \(0=x - 5\), \(x = 5\). ## Step3: Split the integral based on x - intercepts and intervals We need to split the interval \([-4,6]\) into sub - intervals based on the x - intercepts and the behavior of the function. We split it into \([-4,1]\), \([1,3]\), \([3,5]\) and \([5,6]\). The area \(A=\int_{-4}^{1}(-(-x + 1))dx+\int_{1}^{3}(-x + 1)dx+\int_{3}^{5}(-(x - 5))dx+\int_{5}^{6}(x - 5)dx\). ## Step4: Calculate each integral ### Integral 1: \(\int_{-4}^{1}(x - 1)dx=\left[\frac{x^{2}}{2}-x\right]_{-4}^{1}=\left(\frac{1^{2}}{2}-1\right)-\left(\frac{(-4)^{2}}{2}-(-4)\right)=\left(\frac{1}{2}-1\right)-\left(8 + 4\right)=\frac{1 - 2}{2}-12=-\frac{1}{2}-12=-\frac{25}{2}\). But we take the absolute - value, so \(A_1=\frac{25}{2}\). ### Integral 2: \(\int_{1}^{3}(-x + 1)dx=\left[-\frac{x^{2}}{2}+x\right]_{1}^{3}=\left(-\frac{3^{2}}{2}+3\right)-\left(-\frac{1^{2}}{2}+1\right)=\left(-\frac{9}{2}+3\right)-\left(-\frac{1}{2}+1\right)=\left(-\frac{9 - 6}{2}\right)-\left(\frac{-1 + 2}{2}\right)=-\frac{3}{2}-\frac{1}{2}=-2\). Take the absolute - value, \(A_2 = 2\). ### Integral 3: \(\int_{3}^{5}(-(x - 5))dx=\int_{3}^{5}(-x + 5)dx=\left[-\frac{x^{2}}{2}+5x\right]_{3}^{5}=\left(-\frac{5^{2}}{2}+5\times5\right)-\left(-\frac{3^{2}}{2}+5\times3\right)=\left(-\frac{25}{2}+25\right)-\left(-\frac{9}{2}+15\right)=\frac{-25 + 50}{2}-\frac{-9 + 30}{2}=\frac{25}{2}-\frac{21}{2}=2\). ### Integral 4: \(\int_{5}^{6}(x - 5)dx=\left[\frac{x^{2}}{2}-5x\right]_{5}^{6}=\left(\frac{6^{2}}{2}-5\times6\right)-\left(\frac{5^{2}}{2}-5\times5\right)=\left(18-30\right)-\left(\frac{25}{2}-25\right)=-12-\left(\frac{25 - 50}{2}\right)=-12+\frac{25}{2}=\frac{-24 + 25}{2}=\frac{1}{2}\). ## Step5: Sum up the areas \(A=\frac{25}{2}+2 + 2+\frac{1}{2}=\frac{25 + 4+4 + 1}{2}=\frac{34}{2}=17\). # Answer: 17