Lösung 2.1:1d

Aus Online Mathematik Brückenkurs 2

The modulus function, x, strips x of its sign, e.g.

−5=5, 3=3 and −=

For positive values of x, the modulus function has no effect, since x=x, but for negative x the modulus function changes the sign of x, i.e. −x=x (remember that x is negative and therefore −x is positive).

If we draw a graph of y=x it will consist of two parts. For x0 we have y=x, and for x0 we have y=−x

The value of the integral is the area of the region under the graph y=x and between x=−1 and x=2.

This region consists of two triangles and we therefore obtain

\displaystyle \int\limits_{-1}^{2}{\left| x \right|}\,dx=\frac{1}{2}\centerdot 1\centerdot 1+\frac{1}{2}\centerdot 2\centerdot 2=\frac{5}{2}