The value of 12_0^3|x^2-3x+2|,dx is Answer: 22

Question

The value of $12\displaystyle\int_0^3|x^2-3x+2|\,dx$ is

Answer: 22

Accepted Answer

Write $x^2-3x+2=\left(x-\dfrac32\right)^2-\dfrac14$. With $t=x-\dfrac32$:

$$12\int_0^3|x^2-3x+2|\,dx = 24\int_0^{3/2}\left|t^2-\frac14\right|dt = 24\left[\int_0^{1/2}\left(\frac14-t^2\right)dt+\int_{1/2}^{3/2}\left(t^2-\frac14\right)dt\right] = 22$$

Solutions · 1