Editing Integral (section)

==Interpretations==
[[File:Integral approximations J.svg|thumb|Approximations to integral of {{Math|{{radic|''x''}}}} from 0 to 1, with 5 yellow right endpoint partitions and 10 green left endpoint partitions]]
Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many [[infinitesimal]] pieces, then sum the pieces to achieve an accurate approximation.

As another example, to find the area of the region bounded by the graph of the function {{Math|1=''f''(''x'') =}} <math display="inline">\sqrt{x}</math> between {{Math|1=''x'' = 0}} and {{Math|1=''x'' = 1}}, one can divide the interval into five pieces ({{Math|0, 1/5, 2/5, ..., 1}}), then construct rectangles using the right end height of each piece (thus {{Math|{{radic|0}}, {{radic|1/5}}, {{radic|2/5}}, ..., {{radic|1}}}}) and sum their areas to get the approximation

:<math>\textstyle \sqrt{\frac{1}{5}}\left(\frac{1}{5}-0\right)+\sqrt{\frac{2}{5}}\left(\frac{2}{5}-\frac{1}{5}\right)+\cdots+\sqrt{\frac{5}{5}}\left(\frac{5}{5}-\frac{4}{5}\right)\approx 0.7497,</math>
which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, {{Math|2/3}}). One writes

:<math>\int_{0}^{1} \sqrt{x} \,dx = \frac{2}{3},</math>
which means {{Math|2/3}} is the result of a weighted sum of function values, {{Math|1={{radic|''x''}}}}, multiplied by the infinitesimal step widths, denoted by {{Math|''dx''}}, on the interval {{Math|1=[0, 1]}}.

{{multiple image
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<!-- Extra parameters -->| header = Darboux sums
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| image1 = Riemann Integration and Darboux Upper Sums.gif
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| caption1 = Darboux upper sums of the function {{math|''y'' {{=}} ''x''<sup>2</sup>}}
| alt1 = Upper Darboux sum example
| image2 = Riemann Integration and Darboux Lower Sums.gif
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| caption2 = Darboux lower sums of the function {{math|''y'' {{=}} ''x''<sup>2</sup>}}
| alt2 = Lower Darboux sum example
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