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

Orta nokta kuralı, bir eğri altındaki alanı tahmin etmeye yardımcı olur; aralığı eşit alt aralıklara böler.Her alt aralığın ortası, iki uç noktasının ortalaması olarak hesaplanır. Fonksiyonun bu orta noktalardaki değeri, üzerlerine çizilen dikdörtgenlerin yüksekliğini verir. Bu yaklaşım, sol veya sağ uç noktalarını kullanan yöntemlere göre genellikle daha doğru bir tahmin sunar; çünkü alanı az veya fazla tahmin etmekten kaynaklanan hataları azaltır.Bir nehir kenarındaki bir bölgeyi düşünün. Yanlarda iki dikey çizgi, aşağıda yatay bir çizgi ve yukarıda kıvrımlı bir nehir kıyısıyla sınırlandırılmıştır.Alanını tahmin etmek için bölge, eşit genişlikte birkaç dikdörtgene bölünür.Her dikdörtgenin yüksekliği, tabanının ortasındaki fonksiyon kullanılarak bulunur.Her dikdörtgenin alanı, yüksekliği alt aralık genişliğiyle çarpılarak bulunur. Bu alanlar, eğrin altındaki toplam alanı yaklaşık olarak ekleyerek Orta Nokta kural formülü elde edilir.Dikdörtgenlerin sayısı arttıkça ve daraldıkça, tahmin gelişir ve gerçek alana yaklaşır.

ortanokta kuralı

E&#x11F;ri s&#x131;n&#x131;rlar&#x131;n alt&#x131;ndaki alanlar&#x131;n yakla&#x15F;&#x131;k olarak hesaplanmas&#x131;, &#xF6;zellikle kesin bir hesab&#x131;n zor veya pratik olmad&#x131;&#x11F;&#x131; durumlarda, uygulamal&#x131; matematikte yayg&#x131;n bir problemdir. Bu ama&#xE7;la kullan&#x131;lan etkili say&#x131;sal y&#xF6;ntemlerden biri, belirli bir aral&#x131;k &#xFC;zerinde dikd&#xF6;rtgen yakla&#x15F;t&#x131;rmalar kullanarak bir e&#x11F;rinin alt&#x131;ndaki alan&#x131; tahmin eden Ortanokta Kural&#x131;&#x2019;d&#x131;r.
Ortanokta Kural&#x131;n&#x131;n Tan&#x131;m&#x131;
Ortanokta Kural&#x131;, verilen aral&#x131;&#x11F;&#x131; belirli say&#x131;da e&#x15F;it alt aral&#x131;&#x11F;a b&#xF6;lerek ba&#x15F;lar. Her alt aral&#x131;k i&#xE7;in orta nokta, iki u&#xE7; noktan&#x131;n de&#x11F;erlerinin ortalamas&#x131; al&#x131;narak belirlenir. Ard&#x131;ndan, bu orta noktada hesaplanan fonksiyonun de&#x11F;eri, alt aral&#x131;k &#xFC;zerinde olu&#x15F;turulan dikd&#xF6;rtgenin y&#xFC;ksekli&#x11F;ini belirlemek i&#xE7;in kullan&#x131;l&#x131;r. Orta nokta genellikle o aral&#x131;ktaki fonksiyonun ortalama davran&#x131;&#x15F;&#x131;n&#x131; daha iyi yans&#x131;tt&#x131;&#x11F;&#x131;ndan, bu y&#xF6;ntem sol veya sa&#x11F; u&#xE7; noktalar&#x131;n kullan&#x131;lmas&#x131;yla ortaya &#xE7;&#x131;kabilen fazla ya da eksik tahmin hatas&#x131;n&#x131; azaltma e&#x11F;ilimindedir.
Geometrik Yorum
Ortanokta Kural&#x131;na ili&#x15F;kin uygulamal&#x131; bir &#xF6;rnek, bir nehre biti&#x15F;ik bir b&#xF6;lgenin alan&#x131;n&#x131;n tahmin edilmesiyle g&#xF6;sterilebilir. B&#xF6;lge; solda ve sa&#x11F;da dikey do&#x11F;rularla, altta yatay bir do&#x11F;ruyla ve &#xFC;stte e&#x11F;ri bir nehir k&#x131;y&#x131;s&#x131;yla s&#x131;n&#x131;rlanm&#x131;&#x15F;t&#x131;r. &#xDC;st s&#x131;n&#x131;r d&#xFC;zensiz oldu&#x11F;undan, alan basit geometrik form&#xFC;llerle kolayca hesaplanamaz.
Alan Yakla&#x15F;t&#x131;rma S&#xFC;reci
Bu b&#xF6;lgenin alan&#x131;n&#x131; yakla&#x15F;&#x131;k olarak hesaplamak i&#xE7;in b&#xF6;lge, e&#x15F;it geni&#x15F;likte birka&#xE7; dikd&#xF6;rtgene ayr&#x131;l&#x131;r. Her dikd&#xF6;rtgenin y&#xFC;ksekli&#x11F;i, dikd&#xF6;rtgenin taban&#x131;n&#x131;n orta noktas&#x131;nda s&#x131;n&#x131;r e&#x11F;risi de&#x11F;erlendirilerek belirlenir. Her dikd&#xF6;rtgenin alan&#x131;, y&#xFC;ksekli&#x11F;inin geni&#x15F;li&#x11F;iyle &#xE7;arp&#x131;lmas&#x131;yla elde edilir. T&#xFC;m dikd&#xF6;rtgenlerin alanlar&#x131;n&#x131;n toplanmas&#x131;, e&#x11F;ri alt&#x131;ndaki toplam alan i&#xE7;in bir tahmin de&#x11F;eri verir; bu da Ortanokta Kural&#x131;n&#x131;n &#xF6;z&#xFC;n&#xFC; olu&#x15F;turur.
Y&#xF6;ntemin Do&#x11F;rulu&#x11F;u
Dikd&#xF6;rtgen say&#x131;s&#x131; artt&#x131;k&#xE7;a ve geni&#x15F;likleri azald&#x131;k&#xE7;a, dikd&#xF6;rtgen yakla&#x15F;&#x131;m&#x131; e&#x11F;rinin &#x15F;eklini daha yak&#x131;ndan izler. Sonu&#xE7; olarak, tahmin edilen alan iyile&#x15F;ir ve b&#xF6;lgenin ger&#xE7;ek alan&#x131;na yakla&#x15F;&#x131;r.

Ortanokta Kuralı

E&#x11F;ri s&#x131;n&#x131;rlar&#x131;n alt&#x131;ndaki alanlar&#x131;n yakla&#x15F;&#x131;k olarak hesaplanmas&#x131;, &#xF6;zellikle kesin bir ...

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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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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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