# Thread: Integration of absolute value function.. please help

1. ## Integration of absolute value function.. please help

Let f(a) = S02|x(x-a)| dx for 0< a <

(1) Find the function f(a).

(2) Find the minimum of f(a).

Thank you very much for your help!

2. ## Re: Integration of absolute value function.. please help

Originally Posted by kokoro123

Let f(a) = S02|x(x-a)| dx for 0< a <

(1) Find the function f(a).

(2) Find the minimum of f(a).
The integral becomes $\displaystyle \int_0^a {x(a - x)dx} + \int_a^2 {x(x - a)dx}$

3. ## Re: Integration of absolute value function.. please help

Originally Posted by kokoro123
(1) Find the function f(a).
Split it up:

$\displaystyle \int_0^2\left|x(x-a)\right|\,dx$

$\displaystyle =\int_0^a\left|x(x-a)\right|\,dx + \int_a^2\left|x(x-a)\right|\,dx$

$\displaystyle =\int_0^a\left[-x(x-a)\right]\,dx + \int_a^2x(x-a)\,dx$

Originally Posted by kokoro123
(2) Find the minimum of f(a).
Use the Fundamental Theorem of Calculus to differentiate $\displaystyle f$. Locate critical values, etc.