what is the area of the region between the graphs of f(x)=x² - 3x and g(x)=2x from x = 0 to x = 5? choose 1…

Explanation:

Step1: Find the difference function

The difference between the two - functions is $h(x)=g(x)-f(x)=2x-(x^{2}-3x)= - x^{2}+5x$.

Step2: Use the definite - integral formula for area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$ is given by $A=\int_{a}^{b}|g(x)-f(x)|dx$. Here, $a = 0$, $b = 5$, and $h(x)=-x^{2}+5x$. So, $A=\int_{0}^{5}(-x^{2}+5x)dx$.

Step3: Integrate term - by - term

We know that $\int(-x^{2}+5x)dx=-\frac{1}{3}x^{3}+\frac{5}{2}x^{2}+C$.

Step4: Evaluate the definite integral

$A=\left[-\frac{1}{3}x^{3}+\frac{5}{2}x^{2}\right]_{0}^{5}=-\frac{1}{3}(5)^{3}+\frac{5}{2}(5)^{2}-(-\frac{1}{3}(0)^{3}+\frac{5}{2}(0)^{2})$. $A=-\frac{125}{3}+\frac{125}{2}=\frac{-250 + 375}{6}=\frac{125}{6}$.

Answer:

B. $\frac{125}{6}$