### About

#
**Riemann integral**

In two dimensions the **Rieman integral** determines
the area between the *x* axis and the function *y = f(x)* in
an interval *x _{1} < x < x_{2} *by nesting it
between two approximative sums. Both are constructed by a series of
rectangles with intervals 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 smalles value (

*infimum*). The Riemann integral exists, when both sums converge with decreasing interval width, and when they converge to the same limit.

The definition is not identical to the classical *rectangle
algorithm*, where the value of the function is equal to the value at
the beginning of the interval (more generally: always at the same point
in the interval). For "well behaved" functions there will be no
difference in results, yet the Riemann definition is more generally
applicable.

The approximative calculation of the Riemann integral is
shown for the example of a sine function (blue). Its definite integral
is to be calculated in the range *x _{2} - x_{1}*.
The initial value

*x*is defined by a

_{1}**slider**, the end point

*x*by

_{2}**drawing**the red point with the mouse. The yellow curve is the

**analytic solution**

*cos x - cos x*

_{1}.

The number of intervals *(n - 1)* is defined by **slider
n**

*.*defines

**Reset***1 < x < 4*and

*n = 10 (9*intervals

*) .*

The **left window** shows in red the approximation by the supremum
series, with the blue point as the sum; the function always lies below
the rectangle. The **right window** shows the approximation by the
infimum series; the function always lies above the rectangle.

Correspondingly the upper sum of the rectangles is always higher than
the analytic integral, while the lower sum is always lower - for finite
interval widths. With decreasing interval widths both converge to the
same value, the** ***Riemann integral.*

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

Verify that both graphs really limit the intervals in the *y*
direction by the highest, respectively the lowest value in the interval
(observe the sign of the function itself!). Consider the systematic
deviation of the sum from the analytic solution.

**E2: **Compare the construction with the classical *rectangle
(step) algorithm*.** **

**E4:** Increase *n* and observe the process of convergence to
the limit value as the interval width decreases.

**E4: **Draw the end point at a large number for *n* and observe
how the intervals limit lines "paint" the area under the curve. You will
not visibly recognize a difference between sum and integral.

**E5: **Change the initial point with the slider. Draw the end point
beyond the initial point and control if the result is still correct.

### For Teachers

### Translations

### 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

### end faq

### Sample Learning Goals

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### For Teachers

## Riemann Integral JavaScript Simulation Applet HTML5

### Instructions

#### N Slider

#### Draggable Box

#### Toggling Full Screen

#### Reset Button

Research

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### Video

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### Other Resources

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