### About

# Definite integral

This simulation demonstrates definite integration of the sine function by the simple algorithm of summing approximative rectangles. The red curve shows the sine function itself.

The definite integral is to be calculated between initial point *x _{1
}*(blue) an end point

*x*(magenta). A

_{2}**first slider**defines

*x*;

_{1 }*x*can be drawn with the mouse.

_{2}
The red curve is the **analytic antiderivative **for an initial value *x _{1}:
y = cos(x) - cos (x1)*.

A **second slider** *n* defines the number *n-1* of
subinterval into which *x _{2} - x_{1}* is divided
for the approximation. In each subinterval the approximative amplitude
is assumed to be equal to its initial value. The sum of the area of all
rectangles is shown by a green point at

*x*.

_{2}

**Reset **defines 1 < x < 4 and n-1 = 9.

**E1:** Start with the default setting: *x1 = 0; n = 10*.

Draw the magenta end point and observe the deviation of the rectangle sum (green point) from the analytic solution.

**E2: **Understand why the deviation is negative for x **< **π/2,
and why it becomes zero at x = π . Reflect how summing mistakes can
compensate for periodic functions.

**E3: **Increase the number of intervals with the slider and observe
how the deviation changes and disappears in the limit.

**E4: **Change the initial point and consider why and how this shifts
the analytic solution.

**E5: **Choose a large integration interval and consider why the
rectangles lie below or above certain parts of the function.

### For Teachers

### Translations

Code | Language | Translator | Run | |
---|---|---|---|---|

### 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

## Definite 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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