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Lösung 2.1:1d

Aus Online Mathematik Brückenkurs 2

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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The modulus function, <math>|x|</math>, strips <math>x</math> of its sign, e.g.
The modulus function, <math>|x|</math>, strips <math>x</math> of its sign, e.g.
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{{Displayed math||<math>|-5|=5\,</math>, <math>\quad|3|=3\quad</math> and <math>\quad |-\pi|=\pi\,</math>.}}
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{{Abgesetzte Formel||<math>|-5|=5\,</math>, <math>\quad|3|=3\quad</math> and <math>\quad |-\pi|=\pi\,</math>.}}
For positive values of <math>x</math>, the modulus function has no effect, since
For positive values of <math>x</math>, the modulus function has no effect, since
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This region consists of two triangles and we therefore obtain
This region consists of two triangles and we therefore obtain
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{{Displayed math||<math>\int\limits_{-1}^{2} |x|\,dx = \frac{1}{2}\cdot 1\cdot 1 + \frac{1}{2}\cdot 2\cdot 2 = \frac{5}{2}\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>\int\limits_{-1}^{2} |x|\,dx = \frac{1}{2}\cdot 1\cdot 1 + \frac{1}{2}\cdot 2\cdot 2 = \frac{5}{2}\,\textrm{.}</math>}}

Version vom 12:57, 10. Mär. 2009

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

2−1xdx=2111+2122=25. 
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.1:1d“