### About

#
**Riemann integral and Lebesgue integral**

In two dimensions the **Rieman integral **determines
the area between the x axis and the function *y = f(x)* by nesting
it between two approximation sums. Both are constructed by a series of
rectangles with intervals of the integration range along the x axis. For
the *upper sum* the approximative value of *y* in each
interval is equal to the largest value in the interval (its supremum);
for the** ***lower sum*** **it is equal to the smallest value
(infimum). The Riemann integral exists when both sums converge with
decreasing interval width, and when they converge to the same limit. In
this case it also exists for any value in the interval.

The** Lebesgue întegral **divides the integration
area into stripes (not necessarily of the same width) parallel to the x
axis (intervals along the y axis). The size of the area of each interval
is characterized by a measure (taken parallel to the x axis), called the **Lebesgue
measure**. If each stripe has a unique Lebesgue measure, the sum of
the measures is the Lebesgue Integral.

The simulation uses as example a parabola *y = x *^{2}.
The left window shows a *Riemann infimum* series, the right window
a *Lebesgue *series, where the height of the first stripe is
different from that of the other ones. The red line is the Lebesgue
measure. It cuts the function at such a point that the area of the
rectangle is equal to that of the stripe (which has a curved end in *x*
direction). The spandrels to the left and to the right of the vertical
red line have equal area.

The blue curve shows the parabola, whose antiderivative is to be
calculated for an initial value *x _{1} . *The end point

*x*of the integration range (magenta) can be drawn with the mouse. The yellow curve is the analytic solution

_{2}*y*= (

*x*.

^{3}- x_{1 }^{3}) / 3
A first slider defines the initial point *x _{1}*, a second
slider the number of intervals

*n-1*(

*n*in the Lebesgue case).

*Re*se

*t*defines the integration range as

*1 < x < 4*and

*n = 10.*

In the Riemann construction the approximation gets better and better
with increasing *n*. With the special definition of the Lebesgue
measure chosen as an example, the coincidence is perfect independent of
the number of intervals (the measure was calculated analytically; it
could also be calculated in a limiting series process). The concept of
the Lebesgue integral does not itself furnish a special method of
calculation of its measures.

The strength of the Lebesgue concept lies in the very general
mathematical applicability to the concept of integration. It can be
applied to any set of objects, for which a measure is defined. A
function that is Riemann integrable is __always__ Lebesgue
integrable, but not vice versa.

**E1:** Start with default settings: *x _{1} = 1; x_{2 }=
4; n = 10.*

Verify that in the Riemann construction the intervals are closed by the *infimum*
in the *y*-direction. Compare this to the Lebesgue construction.
Consider the systematic deviations from the analytic value.

**E2: **Compare the Riemann lower sum construction with the numerical
rectangle algorithm, starting at the initial point of the interval. Why
are both identical in the parabola example, while they would be
different for the sine function? (the parabola increases monotonically,
while the sine oscillates).

**E3: **Increase the number of intervals and observe the Riemann
lower sum approaching the analytic solution. What is the difference from
the Lebesgue approximation (for the definition of the Lebesgue measure
chosen in this example)?

### For Teachers

### Translations

Code | Language | Translator | Run | |
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### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH-Foundation; Fremont Teng; Loo Kang Wee

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### Sample Learning Goals

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### For Teachers

## Riemann Integral and Lebesgue Integral JavaScript Simulation Applet HTML5

### Instructions

#### n Slider

#### Draggable Boxes

#### Toggling Full Screen

#### Reset Button

Research

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### Video

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### Version:

### Other Resources

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