Name: definite integrals with antiderivatives
Uploaded: 2026-01-12T12:24:04.000Z
Duration: 1 min 29 s
Channel: JoVE

Temel Kalkülüs Teoremi'nin ikinci bölümü, antitürevleri belirli integrallerle bağlar. Belirli bir aralık boyunca bir fonksiyonun hızının tam toplam birikimini bulur.Bir fonksiyon kapalı aralıkta sürekliyse, belirli integral antitürevin üst sınırdaki değerinden en düşük sınırdaki değerine eşittir.Bu, eğriyi alt altındaki birçok dar dikdörtgen kullanarak alanı tahmin eden Riemann toplam yaklaşımlarından kaçınır.Seçilen antitürev nihai sonucu etkilemez. Örneğin, x'in G'si, başka bir antitürevatten sabit bir C ile farklı olabilir, ancak C, aralığın uç noktalarındaki değerleri çıkardığında iptal eder.Yaygın bir uygulama, bir nesnenin net yer değiştirmesini hız fonksiyonu kullanarak bulmaktır.Belirli bir zaman aralığı için net yer değiştirme, 1 saniyeden 4 saniyeye kadar değişir ve hız-zaman grafiğinin altındaki alandır.Bunu hesaplamak için önce hız fonksiyonunun antitürevini bulun—bu pozisyon fonksiyonunu verir. Sonra, teoremi kullanarak aralığın uç noktalarındaki konumu değerlendirin. Konum fonksiyonunun 1 saniyelik değerini, 4 saniyelik değerinden çıkararak net yer değiştirmeyi elde edersiniz.

kalkülüsün temel teoremi ii

Kalk&#xFC;l&#xFC;ste, s&#xFC;rekli bir e&#x11F;rinin alt&#x131;ndaki alan&#x131;n hesaplanmas&#x131;, Kalk&#xFC;l&#xFC;s&#xFC;n Temel Teoremi II&#x2019;nin uygulanmas&#x131;yla temelde basitle&#x15F;tirilmi&#x15F;tir. Sonsuz say&#x131;da sonsuz k&#xFC;&#xE7;&#xFC;k dikd&#xF6;rtgenin toplanmas&#x131;na dayanan limit s&#xFC;recine ba&#x15F;vurmak yerine, bu teorem antit&#xFC;revler arac&#x131;l&#x131;&#x11F;&#x131;yla do&#x11F;rudan hesaplamaya olanak tan&#x131;r ve b&#xF6;ylece belirli integral hesaplama s&#xFC;recini kolayla&#x15F;t&#x131;r&#x131;r.Kalk&#xFC;l&#xFC;s&#xFC;n Temel Teoremi II, f(x) fonksiyonu [a, b] kapal&#x131; aral&#x131;&#x11F;&#x131;nda s&#xFC;rekli ise, f(x)&#x2019;in bu aral&#x131;k &#xFC;zerindeki belirli integralinin antit&#xFC;revlerinden biri F(x) kullan&#x131;larak hesaplanabilece&#x11F;ini s&#xF6;yler. Matematiksel olarak &#x15F;u &#x15F;ekilde ifade edilir:\begin{equation*}\int_a^b f(x)\,dx = F(b) - F(a)\end{equation*}Bu sonu&#xE7;, bir fonksiyonun t&#xFC;m antit&#xFC;revlerinin yaln&#x131;zca bir sabit terim kadar birbirinden farkl&#x131; olmas&#x131; ger&#xE7;e&#x11F;ine dayan&#x131;r. Bu sabit, F(b)&#x2212;F(a) fark&#x131; hesaplan&#x131;rken sadele&#x15F;ti&#x11F;inden, herhangi bir antit&#xFC;rev integralin hesaplanmas&#x131;nda kullan&#x131;labilir; bu durum, hesaplama esnekli&#x11F;ini b&#xFC;y&#xFC;k &#xF6;l&#xE7;&#xFC;de art&#x131;r&#x131;r.f(x) fonksiyonunun [a, b] aral&#x131;&#x11F;&#x131;n&#x131;n tamam&#x131;nda s&#xFC;rekli olmas&#x131; &#xE7;ok &#xF6;nemlidir. Aral&#x131;k i&#xE7;inde s&#xFC;reksizliklerin bulunmas&#x131; veya fonksiyonun tan&#x131;ms&#x131;z oldu&#x11F;u noktalar&#x131;n bulunmas&#x131;, teoremin uygulanabilirli&#x11F;ini ortadan kald&#x131;r&#x131;r. Bu gibi durumlarda, belirli integral bu y&#xF6;ntemle do&#x11F;ru tan&#x131;ml&#x131; olmayabilir; alternatif tekniklerin kullan&#x131;lmas&#x131; veya par&#xE7;al&#x131; integrasyonun dikkatle uygulanmas&#x131; gerekir.Bu teorem, integrali sonsuz k&#xFC;&#xE7;&#xFC;k niceliklerin toplanmas&#x131;na dayal&#x131; bir s&#xFC;re&#xE7; olmaktan &#xE7;&#x131;kararak, belirli u&#xE7; noktalardaki fonksiyon de&#x11F;erlerinin de&#x11F;erlendirilmesine dayal&#x131; bir yakla&#x15F;&#x131;ma d&#xF6;n&#xFC;&#x15F;t&#xFC;r&#xFC;r. Matematik, fizik ve m&#xFC;hendislikteki &#xE7;e&#x15F;itli problemlere analitik &#xE7;&#xF6;z&#xFC;mler sa&#x11F;lamada integral kalk&#xFC;l&#xFC;s&#xFC;n&#xFC;n temel ta&#x15F;lar&#x131;ndan birini olu&#x15F;turur.

Kalkülüsün Temel Teoremi II

Kalk&#xFC;l&#xFC;ste, s&#xFC;rekli bir e&#x11F;rinin alt&#x131;ndaki alan&#x131;n hesaplanmas&#x131;, Kalk&#xFC;l&#xFC;s&#xFC;n Temel Teoremi ...

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JoVE Core

Calculus

Kalkülüs

Unit

Birim

Integrals

İntegraller

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

Fundamental Theorem of Calculus II

definite integrals with antiderivatives

The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative.

If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because any two antiderivatives of the same function differ only by a constant.

That constant does not change the final answer. When F(b) - F(a) is calculated, the constant cancels out. This gives flexibility, because any antiderivative can be used to find the integral.

The theorem only applies when the function is continuous on the entire interval [a, b]. If the function has a discontinuity or is undefined somewhere inside the interval, this method does not work. In those cases, the definite integral may need another approach, such as careful piecewise integration.

This theorem turns integration into a process based on endpoint values instead of infinite sums. It is a key idea in integral calculus and helps solve problems in mathematics, physics, and engineering.

Definite Integrals with Antiderivatives

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key terms and concepts

Learn how the Fundamental Theorem of Calculus links definite integrals and antiderivatives, and why continuity on [a, b] matters.

Definite Integrals with Antiderivatives | 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

Definite Integrals with Antiderivatives

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