Question

Define a function $f:R\to R$ by $f\left(x\right)=max\{|x|,|x-1|,....|x-2n|\}$
Where $n$ is a fixed natural number, Then ${\int }_0^{2n}f\left(x\right)dx$ is.

• A. $n$
• B. $n^2$
• C. $3n$
• D. $3n^2$

Answer 1

The correct option is D $3n^2$

${\int }_0^{2n}f\left(x\right)dx={\int }_0^n|x-2n|dx+{\int }_n^{2n}|x|dx$
${\int }_0^n(2n-x)dx+{\int }_n^{2n}xdx$
$={(2nx-\frac{x^2}{2})}_0^n+{\left(\frac{x^2}{2}\right)}_0^{2n}=3n^2$.