Aus Online Mathematik Brückenkurs 2

Version vom 12:09, 21. Okt. 2008 (bearbeiten) Tek (Diskussion | Beiträge) K ← Zum vorherigen Versionsunterschied		Version vom 12:57, 10. Mär. 2009 (bearbeiten) (rückgängig) Tekbot (Diskussion | Beiträge) K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)) Zum nächsten Versionsunterschied →
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	The value of the integral is		The value of the integral is
-	{{Displayed math||<math>\begin{align}	+	{{Abgesetzte Formel||<math>\begin{align}
	\int\limits_{0}^{1} (2x+1)\,dx		\int\limits_{0}^{1} (2x+1)\,dx
	&= \text{(area of the square)} + \text{(area of the triangle)}\\		&= \text{(area of the square)} + \text{(area of the triangle)}\\
	&= 1\cdot 1 + \frac{1}{2}\cdot 1\cdot 2 = 2\,\textrm{.}		&= 1\cdot 1 + \frac{1}{2}\cdot 1\cdot 2 = 2\,\textrm{.}
	\end{align}</math>}}		\end{align}</math>}}

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Version vom 12:57, 10. Mär. 2009

The graph of the function \displaystyle y=2x+1 is a straight line which cuts the y-axis at \displaystyle y=1 and has slope 2.

The integral's value is the area under the straight line and between \displaystyle x=0 and \displaystyle x=1.

We can divide up the region under the graph into a square and rectangle,

and then add up the area to obtain the total area.

The value of the integral is

\displaystyle \begin{align} \int\limits_{0}^{1} (2x+1)\,dx &= \text{(area of the square)} + \text{(area of the triangle)}\\ &= 1\cdot 1 + \frac{1}{2}\cdot 1\cdot 2 = 2\,\textrm{.} \end{align}