MHB Compute $\int_{0}^{t}|s-t|\,ds$: Step-by-Step Guide

[Fermat1]

How would I compute the following integral? Let t be in [0,1].

$\int_{0}^{t}|s-t|\,ds$

 

[alyafey22]

How would I compute the following integral? Let t be in [0,1].

$\int_{0}^{t}|s-t|\,ds$

Since you are integrating on the interval $[0,t]$ so we have $0\leq s \leq t$.

 

[HallsofIvy]

How would I compute the following integral? Let t be in [0,1].

$\int_{0}^{t}|s-t|\,ds$

The integral is for s from 0 to t so s is always less than t and s- t is always negative. |s- t|= -(s- t) so
\int_0^{t}|s- t| ds= -\int_0^t (t- s) ds= \left[ts- s^2/2\right]_0^t= t^2/2