Find the integral of the absolute value function

Show that

$\int_0^x |t| \, dt = \frac{1}{2} x |x|$

for all $x \in \mathbb{R}$ .

Proof. We consider three cases.
Case #1: If $x = 0$ , then both sides of the equation are 0, so the equation holds.
Case #2: If $x > 0$ , then $|t| = t$ for all $t \in [0,x]$ so,

$\int_0^x |t| \, dt = \int_0^x t \, dt = \frac{1}{2}x^2 = \frac{1}{2}x |x|.$

The last equality follows since $x > 0$ implies $x = |x|$ .
Case #3: If $x < 0$ , then $|t| = -t$ for all $\in [x,0]$ so,

$\begin{align*} \int_0^x |t| \, dt &= - \int_0^x t \, dt \\ &= \int_x^0 t \, dt \\ &= \left. \frac{1}{2}t^2 \right|_x^0 \\ &= -\frac{1}{2}x^2 \\ &= \frac{1}{2} x |x|. \end{align*}$

The final line follows since $|x| = -x$ since $x < 0. \qquad \blacksquare$

Stumbling Robot

Find the integral of the absolute value function

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