evaluate the following integral by interpreting it in terms of area. ∫_{-3}^{7}(|x - 2| - 3)dx submit your…

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)