let r be the region bounded by the functions f(x)=3x² - 5x and g(x)=x as shown in the diagram below. find…

Explanation:

Step1: Find intersection points

Set $f(x)=g(x)$, so $3x^{2}-5x = x$. Rearrange to $3x^{2}-6x = 0$, factor out $3x$: $3x(x - 2)=0$. The solutions are $x = 0$ and $x = 2$.

Step2: Determine upper - lower function

On the interval $[0,2]$, $g(x)\geq f(x)$. The area $A$ between two curves $y = g(x)$ and $y = f(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}[g(x)-f(x)]dx$. Here, $a = 0$, $b = 2$, $g(x)-f(x)=x-(3x^{2}-5x)=-3x^{2}+6x$.

Step3: Calculate the integral

$\int_{0}^{2}(-3x^{2}+6x)dx=\left[-x^{3}+3x^{2}\right]_{0}^{2}$. Substitute the upper and lower limits: $(-2^{3}+3\times2^{2})-(-0^{3}+3\times0^{2})=(-8 + 12)-0 = 4$.

Answer:

$4$