Name: estimating area with the midpoint rule
Uploaded: 2026-01-12T12:24:26.000Z
Duration: 1 min 20 s
Channel: JoVE

Reguła punktu środkowego pomaga oszacować powierzchnię pod krzywą poprzez podział przedziału na równe podprzedziały.Środek każdego podprzedziału oblicza się jako średnią jego dwóch końcowych punktów. Wartość funkcji w tych punktach środkowych określa wysokość prostokątów narysowanych nad nimi. Takie podejście często daje dokładniejsze oszacowanie niż metody wykorzystujące lewy lub prawy koniec końcowy, ponieważ zmniejsza błędy spowodowane zaniżeniem lub zawyżaniem obszaru.Weźmy pod uwagę region obok rzeki. Jest ograniczone dwoma pionowymi liniami po bokach, poziomą linią poniżej oraz zakrzywionym brzegiem rzeki powyżej.Aby oszacować jego powierzchnię, region dzieli się na kilka prostokątów o równej szerokości.Wysokość każdego prostokąta jest określana za pomocą funkcji znajdującej się w środku jego podstawy.Pole każdego prostokąta określa się, mnożąc jego wysokość przez szerokość podprzedziału. Te obszary są dodawane, aby przybliżyć całkowitą powierzchnię pod krzywą, dając wzór reguły punktu środkowego.Wraz ze wzrostem liczby prostokątów i ich zwężeniem szacunkowa liczba rośnie i zbliża się do rzeczywistej powierzchni.

metoda punktu środkowego

Aproksymacja p&#xF3;l pod zakrzywionymi granicami jest cz&#x119;stym problemem w matematyce stosowanej, szczeg&#xF3;lnie gdy dok&#x142;adne obliczenia s&#x105; trudne lub niepraktyczne. Jedn&#x105; ze skutecznych metod numerycznych w tym celu jest metoda punktu &#x15B;rodkowego, kt&#xF3;ra pozwala oszacowa&#x107; pole pod krzyw&#x105; za pomoc&#x105; prostok&#x105;tnych przybli&#x17C;e&#x144; w okre&#x15B;lonym przedziale.
Opis metody punktu &#x15B;rodkowego
Metoda punktu &#x15B;rodkowego rozpoczyna si&#x119; od podzielenia danego przedzia&#x142;u na kilka r&#xF3;wnych podprzedzia&#x142;&#xF3;w. Dla ka&#x17C;dego podprzedzia&#x142;u punkt &#x15B;rodkowy podprzedzia&#x142;u wyznacza si&#x119; jako &#x15B;redni&#x105; arytmetyczn&#x105; jego dw&#xF3;ch punkt&#xF3;w ko&#x144;cowych. Warto&#x15B;&#x107; funkcji w tym punkcie &#x15B;rodkowym jest nast&#x119;pnie wykorzystywana do okre&#x15B;lenia wysoko&#x15B;ci prostok&#x105;ta skonstruowanego na tym podprzedziale. Poniewa&#x17C; &#x15B;rodek przedzia&#x142;u cz&#x119;sto le&#x17C;y bli&#x17C;ej &#x15B;redniego zachowania warto&#x15B;ci funkcji na tym przedziale, metoda ta zmniejsza b&#x142;&#x105;d zwi&#x105;zany z przeszacowaniem lub niedoszacowaniem, kt&#xF3;ry mo&#x17C;e wyst&#x105;pi&#x107; w przypadku u&#x17C;ycia lewych lub prawych punkt&#xF3;w ko&#x144;cowych.
Interpretacja geometryczna
Praktycznym przyk&#x142;adem metody punktu &#x15B;rodkowego mo&#x17C;e by&#x107; oszacowanie pola obszaru przylegaj&#x105;cego do rzeki. Obszar ten jest ograniczony z lewej i prawej strony prostymi pionowymi, od do&#x142;u prost&#x105; poziom&#x105;, a od g&#xF3;ry zakrzywionym brzegiem rzeki. Poniewa&#x17C; g&#xF3;rna granica jest nieregularna, jej pole nie mo&#x17C;e by&#x107; &#x142;atwo obliczone za pomoc&#x105; prostych wzor&#xF3;w geometrycznych.
Proces przybli&#x17C;ania pola
Aby oszacowa&#x107; pole tego obszaru, dzieli si&#x119; go na kilka prostok&#x105;t&#xF3;w o r&#xF3;wnej szeroko&#x15B;ci. Wysoko&#x15B;&#x107; ka&#x17C;dego prostok&#x105;ta okre&#x15B;la si&#x119;, obliczaj&#x105;c warto&#x15B;&#x107; funkcji w punkcie &#x15B;rodkowym podstawy prostok&#x105;ta. Pole ka&#x17C;dego prostok&#x105;ta oblicza si&#x119;, mno&#x17C;&#x105;c jego wysoko&#x15B;&#x107; przez szeroko&#x15B;&#x107;. Zsumowanie p&#xF3;l wszystkich prostok&#x105;t&#xF3;w daje oszacowanie ca&#x142;kowitego pola pod krzyw&#x105;, co stanowi istot&#x119; metody punktu &#x15B;rodkowego.
Dok&#x142;adno&#x15B;&#x107; metody
Wraz ze wzrostem liczby prostok&#x105;t&#xF3;w i zmniejszeniem ich szeroko&#x15B;ci przybli&#x17C;enie prostok&#x105;tne coraz dok&#x142;adniej odzwierciedla kszta&#x142;t krzywej. W rezultacie szacowane pole poprawia si&#x119; i zbli&#x17C;a do rzeczywistego pola obszaru.

Metoda punktu środkowego

Aproksymacja p&#xF3;l pod zakrzywionymi granicami jest cz&#x119;stym problemem w matematyce stosowanej, szczeg&#xF3;lnie gdy dok&#x142;adne obliczenia ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Integrals

Całki

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Midpoint Rule

estimating area with the midpoint rule

The Midpoint Rule estimates the area under a curve by using rectangles. It is a numerical method that helps when an exact area is hard to find or not practical to calculate. By using rectangle heights from midpoints, it gives a useful approximation over a chosen interval.

To use the Midpoint Rule, the interval is split into equal subintervals. The midpoint of each subinterval is found by averaging its two endpoints. The function value at that midpoint gives the height of a rectangle over that subinterval.

This midpoint choice often matches the average behavior of the curve better than using the left or right endpoint. That can reduce the error that may happen with overestimation or underestimation. In this way, the method gives a closer area estimate than simple endpoint-based rectangles in many cases.

The idea can be shown with a region next to a river. The region is bounded on the left and right by vertical lines, below by a horizontal line, and above by a curved riverbank. Since the upper edge is irregular, its area is not easy to find with simple geometry.

The area is approximated by dividing the region into several rectangles of equal width. Each rectangle’s area is found by multiplying its width by its midpoint height. Adding the areas of all the rectangles gives the estimated total area under the curve.

The estimate improves as the number of rectangles increases and their widths get smaller. The rectangles then follow the curve more closely. As a result, the approximation approaches the true area of the region.

Estimating Area with the Midpoint Rule

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key terms and concepts

Learn how the Midpoint Rule estimates area under a curve with equal-width rectangles, midpoint heights, and better accuracy as intervals get smaller.

Estimating Area with the Midpoint Rule | 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.

calculus

Estimating Area with the Midpoint Rule

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