let r be the region bounded by the functions f(x)=2x² and g(x)= - 3x² - 10x as shown in the diagram below…

Explanation:

Step1: Find intersection points

Set $f(x)=g(x)$, so $2x^{2}=-3x^{2}-10x$. Rearranging gives $5x^{2}+10x = 0$, factoring out $5x$ we get $5x(x + 2)=0$. The solutions are $x=-2$ and $x = 0$.

Step2: Set up integral for area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$ is $A=\int_{a}^{b}|f(x)-g(x)|dx$. Here $f(x)\geq g(x)$ on $[-2,0]$, so $A=\int_{-2}^{0}(2x^{2}-(-3x^{2}-10x))dx=\int_{-2}^{0}(5x^{2}+10x)dx$.

Step3: Integrate

Using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int(5x^{2}+10x)dx=\frac{5x^{3}}{3}+5x^{2}+C$.

Step4: Evaluate definite integral

$A=\left[\frac{5x^{3}}{3}+5x^{2}\right]_{-2}^0$. First, substitute $x = 0$: $\frac{5(0)^{3}}{3}+5(0)^{2}=0$. Then substitute $x=-2$: $\frac{5(-2)^{3}}{3}+5(-2)^{2}=\frac{-40}{3}+20=\frac{-40 + 60}{3}=\frac{20}{3}\approx6.667$.

Answer:

$6.667$