# integral of modulus

#### Fermat

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

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

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
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

##### Well-known member
MHB Math Helper
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
$$\int_0^{t}|s- t| ds= -\int_0^t (t- s) ds= \left[ts- s^2/2\right]_0^t= t^2/2$$