Animated Solution for Mathematics - Definite Integration: The value of $12{\int }_0^3|x^2-3x+2|dx$ is ______.

We are tasked with evaluating the integral:

The modulus function acts as a filter that ensures the output is always non-negative. To evaluate this, we must analyze the behavior of the quadratic function $f(x)=x^2-3x+2$.

By factorizing the quadratic, we find:

The roots of the function are $x=1$ and $x=2$. These points are critical because they define where the parabola crosses the $x$-axis and where the sign of the expression changes.

As we traverse the interval $[0,3]$, the function $f(x)$ behaves differently in three distinct regions:

1. On $[0,1]$, $f(x)\ge 0$. 2. On $[1,2]$, $f(x)<0$. 3. On $[2,3]$, $f(x)\ge 0$.

Because of the modulus, we must negate the expression in the interval where it is negative to ensure the area remains positive. We split the integral accordingly:

The antiderivative of $f(x)=x^2-3x+2$ is given by:

We evaluate $F(x)$ at the boundaries $0,1,2,$ and $3$: $F(0)=0$ $F(1)=\frac{1}{3}-\frac{3}{2}+2=\frac{5}{6}$ $F(2)=\frac{8}{3}-6+4=\frac{2}{3}$ $F(3)=9-\frac{27}{2}+6=\frac{3}{2}$

Now, we compute the value of each segment: First part: $F(1)-F(0)=\frac{5}{6}$ Second part: $-(F(2)-F(1))=-(\frac{2}{3}-\frac{5}{6})=-(-\frac{1}{6})=\frac{1}{6}$ * Third part: $F(3)-F(2)=\frac{3}{2}-\frac{2}{3}=\frac{5}{6}$

Summing these components, we obtain:

Finally, multiplying by the constant $12$ outside the integral:

The final answer is 22.