find the area of the shaded region.\nthe total area of the shaded regions is \n(type an integer or a…

Explanation:

Step1: Find x - intercepts

Set $y = x^{2}-2x=0$, factor out $x$: $x(x - 2)=0$. So $x = 0$ and $x = 2$ are the x - intercepts.

Step2: Split the integral

The shaded region is composed of two parts. We need to calculate the integral $\int_{- 1}^{0}(x^{2}-2x)dx+\int_{0}^{2}-(x^{2}-2x)dx$.

Step3: Calculate the first integral

Use the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$. $\int_{-1}^{0}(x^{2}-2x)dx=\left[\frac{x^{3}}{3}-x^{2}\right]_{-1}^{0}=(0 - 0)-\left(\frac{(-1)^{3}}{3}-(-1)^{2}\right)=-\left(-\frac{1}{3}-1\right)=\frac{1 + 3}{3}=\frac{4}{3}$.

Step4: Calculate the second integral

$\int_{0}^{2}-(x^{2}-2x)dx=-\int_{0}^{2}(x^{2}-2x)dx=-\left[\frac{x^{3}}{3}-x^{2}\right]_{0}^{2}=-\left(\frac{2^{3}}{3}-2^{2}-(0 - 0)\right)=-\left(\frac{8}{3}-4\right)=-\frac{8 - 12}{3}=\frac{4}{3}$.

Step5: Find the total area

The total area $A=\frac{4}{3}+\frac{4}{3}=\frac{8}{3}$.

Answer:

$\frac{8}{3}$