evaluate the following integral by interpreti...

# Explanation: ## Step1: Analyze the absolute - value function The function \(y = |x - 2|-3\) can be written as a piece - wise function. When \(x-2\geq0\) (i.e., \(x\geq2\)), \(y=(x - 2)-3=x - 5\); when \(x - 2<0\) (i.e., \(x<2\)), \(y=-(x - 2)-3=-x - 1\). ## Step2: Split the integral We split the integral \(\int_{-3}^{7}(|x - 2|-3)dx=\int_{-3}^{2}(-x - 1)dx+\int_{2}^{7}(x - 5)dx\). ## Step3: Calculate the first integral \(\int_{-3}^{2}(-x - 1)dx\) Using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)\), we have \(\int_{-3}^{2}(-x - 1)dx=-\int_{-3}^{2}x dx-\int_{-3}^{2}1dx\). \[ \begin{align*} -\left[\frac{x^{2}}{2}\right]_{-3}^{2}-[x]_{-3}^{2}&=-\left(\frac{2^{2}}{2}-\frac{(-3)^{2}}{2}\right)-(2-( - 3))\\ &=-\left(2-\frac{9}{2}\right)-5\\ &=-\left(\frac{4 - 9}{2}\right)-5\\ &=\frac{5}{2}-5\\ &=-\frac{5}{2} \end{align*} \] ## Step4: Calculate the second integral \(\int_{2}^{7}(x - 5)dx\) \(\int_{2}^{7}(x - 5)dx=\int_{2}^{7}x dx-5\int_{2}^{7}1dx\). \[ \begin{align*} \left[\frac{x^{2}}{2}\right]_{2}^{7}-5[x]_{2}^{7}&=\frac{7^{2}}{2}-\frac{2^{2}}{2}-5(7 - 2)\\ &=\frac{49}{2}-2-25\\ &=\frac{49 - 4}{2}-25\\ &=\frac{45}{2}-25\\ &=\frac{45 - 50}{2}\\ &=-\frac{5}{2} \end{align*} \] ## Step5: Sum the two results \(\int_{-3}^{7}(|x - 2|-3)dx=-\frac{5}{2}-\frac{5}{2}=-5\). # Answer: \(-5\)