Name: riemann sums and area estimates
Uploaded: 2026-01-12T12:24:23.000Z
Duration: 1 min 29 s
Channel: JoVE

Betrachte eine Funktion über einem abgeschlossenen Intervall. Um die Fläche unter der Kurve zu approximieren, wird das Intervall in gleiche Unterintervalle unterteilt, die Rechtecke gleicher Breite bilden.Wählen Sie einen beliebigen Punkt innerhalb des Teilintervalls aus und verwenden Sie dort den Funktionswert als Rechteckshöhe. Multipliziert man die Höhe mit der Breite, erhält man die Fläche dieses Rechtecks. Die Gesamtfläche unter der Kurve wird durch Summieren aller Rechtecke approximiert und wird als Riemannsche Summe bezeichnet.Das definite Integral einer Funktion über einem gegebenen Intervall ist definiert als der Grenzwert ihrer Riemannschen Summe, wenn die Anzahl der Teilintervalle sich dem Unendlichen nähert und die Breite jedes Rechtecks infinitesimal klein wird, vorausgesetzt, dieser Grenzwert existiert.Betrachten wir zum Beispiel ein Auto, das mit variabler Geschwindigkeit unterwegs ist. Teile die gesamte Reisezeit in gleiche Zeitintervalle.In jedem Intervall wählen Sie einen beliebigen Zeitpunkt und nehmen Sie die entsprechende Geschwindigkeit. Multiplizieren Sie die Geschwindigkeit mit der Intervallbreite, um die zurückgelegte Strecke abzuschätzen.Die Summe davon bildet eine Riemannsche Summe, und in diesem Beispiel wird die Fläche unter der Kurve physikalisch als die vom Auto zurückgelegte Strecke interpretiert. Wenn die Geschwindigkeit positiv bleibt, wird der Gesamtabstand durch Integration der Geschwindigkeitsfunktion innerhalb der Grenzen ermittelt.

bestimmtes integral

Betrachtet man eine reellwertige Funktion, die auf einem abgeschlossenen Intervall definiert ist, so besteht eine der zentralen Aufgaben der Analysis darin, die Fl&#xE4;che unter dem Graphen einer solchen Funktion zu bestimmen. Ist eine exakte Berechnung nicht unmittelbar m&#xF6;glich, kann die Fl&#xE4;che durch Unterteilung des Intervalls in eine endliche Anzahl gleich gro&#xDF;er Teilintervalle n&#xE4;herungsweise bestimmt werden. Jedes Teilintervall entspricht einem Rechteck, dessen Breite der L&#xE4;nge des Teilintervalls entspricht und dessen H&#xF6;he durch den Funktionswert an einem gew&#xE4;hlten Punkt innerhalb des Teilintervalls bestimmt wird.
Riemann-Summen und Fl&#xE4;chenn&#xE4;herung
F&#xFC;r jedes Teilintervall wird ein beliebiger Punkt ausgew&#xE4;hlt, und der Funktionswert an diesem Punkt dient als H&#xF6;he des Rechtecks. Die Fl&#xE4;che jedes Rechtecks ergibt sich aus der Multiplikation von H&#xF6;he und Breite. Die Summe der Fl&#xE4;chen aller Rechtecke liefert eine N&#xE4;herung f&#xFC;r die Gesamtfl&#xE4;che unter der Kurve. Diese N&#xE4;herung wird als Riemannsumme bezeichnet. Mit zunehmender Anzahl der Teilintervalle und abnehmender Breite werden die N&#xE4;herungen genauer und spiegeln den Verlauf der Kurve immer pr&#xE4;ziser wider.
Bestimmtes Integral als Grenzwertprozess
Das bestimmte Integral pr&#xE4;zisiert das Konzept der Riemannsumme, indem untersucht wird, was geschieht, wenn die Unterteilung des Intervalls unendlich fein wird. Das bestimmte Integral einer Funktion &#xFC;ber ein gegebenes Intervall ist definiert als der Wert, dem sich die Riemannsummen ann&#xE4;hern, wenn die Anzahl der Rechtecke unbegrenzt w&#xE4;chst und die Breite jedes Rechtecks beliebig klein wird, vorausgesetzt, dieser Grenzwert existiert. In diesem Sinne stellt das bestimmte Integral die exakte Fl&#xE4;che unter der Kurve &#xFC;ber dem betreffenden Intervall dar.
Interpretation in physikalischen Anwendungen
Ein anschauliches Beispiel findet sich in der Bewegungsanalyse. Betrachtet man ein Auto, das auf einer geraden Strecke mit zeitlich variierender Geschwindigkeit f&#xE4;hrt, so l&#xE4;sst sich durch Unterteilung der gesamten Fahrzeit in gleich lange Abschnitte und durch Verwendung der Geschwindigkeit zu einem gew&#xE4;hlten Zeitpunkt innerhalb jedes Abschnitts die in diesem Abschnitt zur&#xFC;ckgelegte Strecke absch&#xE4;tzen. Die Summe dieser Strecken ergibt eine Riemannsumme, die die insgesamt zur&#xFC;ckgelegte Strecke approximiert. Bleibt die Geschwindigkeit &#xFC;ber das gesamte Intervall positiv, entspricht das bestimmte Integral der Geschwindigkeitsfunktion der insgesamt vom Auto zur&#xFC;ckgelegten Strecke.

Bestimmtes Integral

Betrachtet man eine reellwertige Funktion, die auf einem abgeschlossenen Intervall definiert ist, so besteht eine der zentralen Aufgaben der Analysis ...

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riemann sums and area estimates

Riemann sums and area estimates are a key part of calculus for finding the area under a curve. When a real-valued function is defined on a closed interval, the area under its graph can be approximated if an exact calculation is not easy to do.

The interval is split into equal subintervals. Each subinterval becomes a rectangle. The rectangle’s width is the length of the subinterval, and its height comes from the function value at one chosen point inside that part of the interval. Multiplying height by width gives the area of each rectangle.

Adding the areas of all the rectangles gives a Riemann sum. This sum is an approximation of the total area under the curve. As the number of subintervals increases and the widths get smaller, the estimate usually becomes more accurate because it follows the shape of the curve more closely.

The definite integral builds on this idea by taking the limit of the Riemann sums. It looks at what happens as the partition of the interval becomes infinitely fine and the rectangle widths become negligibly small, provided that the limit exists. In that case, the definite integral gives the exact area under the curve over the interval.

This idea also has a physical meaning in motion. A car moving along a straight path with changing velocity can be studied by dividing the travel time into equal parts. Using the velocity at a chosen time in each part gives an estimate for the distance traveled in that segment. Adding those distances gives a Riemann sum for total distance, and when the velocity stays positive, the definite integral of velocity over time matches the total distance covered.

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key terms and concepts

Learn how Riemann sums use rectangles to estimate area under a curve and how the definite integral gives the exact value as the intervals shrink.

Riemann Sums and Area Estimates | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

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Riemann Sums and Area Estimates

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