let r be the region bounded by the functions f(x)=-2x² + 5x and g(x)=2x² - 7x as shown in the diagram below…

Explanation:

Step1: Find intersection points

Set $f(x)=g(x)$, so $-2x^{2}+5x = 2x^{2}-7x$. Rearranging gives $4x^{2}-12x = 0$, factoring out $4x$ we get $4x(x - 3)=0$. The solutions are $x = 0$ and $x=3$.

Step2: Determine upper - lower function

For $0<x<3$, $f(x)-g(x)=(-2x^{2}+5x)-(2x^{2}-7x)=-4x^{2}+12x$.

Step3: Calculate the area

The area $A=\int_{0}^{3}(-4x^{2}+12x)dx$. Using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int(-4x^{2}+12x)dx=-4\times\frac{x^{3}}{3}+12\times\frac{x^{2}}{2}+C=-\frac{4}{3}x^{3}+6x^{2}+C$. Evaluating the definite integral: $\left(-\frac{4}{3}(3)^{3}+6(3)^{2}\right)-\left(-\frac{4}{3}(0)^{3}+6(0)^{2}\right)$. $-\frac{4}{3}\times27 + 6\times9-0=-36 + 54=18$.

Answer:

$18$