Definite Integral with Absolute Value

[Nate River]

Hi there!

I am just struggling with how this author goes from the yellow question mark to the yellow comment box in this proof. I feel like it has something to do with the absolute values that lead to a piecewise function but I am not sure. theta is uniformly distributed.

 

[Dr.Peterson]

Have you tried carrying out the work for each case? We need to see what you've done and where you are stuck.

Take the first case, with [MATH]r_2\le a[/MATH]. Then throughout the interval [a,b], [MATH]r_2 - \theta \le 0[/MATH], so [MATH]|r_2 - \theta| = \theta - r_2[/MATH]. Put that into the integral and see what you get.

The second case will be a little more complicated (you'll need that piecewise function you expected).

 

[Nate River]

Have you tried carrying out the work for each case? We need to see what you've done and where you are stuck.

Take the first case, with [MATH]r_2\le a[/MATH]. Then throughout the interval [a,b], [MATH]r_2 - \theta \le 0[/MATH], so [MATH]|r_2 - \theta| = \theta - r_2[/MATH]. Put that into the integral and see what you get.

The second case will be a little more complicated (you'll need that piecewise function you expected).

I managed to figure it out, thank you for taking the time to respond!