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

Druga część Fundamentalnego twierdzenia rachunku różniczkowego łączy pochodne z całkami określonymi. Określa dokładną całkowitą akumulację szybkości funkcji w określonym przedziale.Jeśli funkcja jest ciągła na przedziale domkniętym, całka określona jest równa wartości pochodnej w górnej granicy minus jej wartość w dolnej granicy.To unika przybliżeń sumy Riemanna, które szacują pole za pomocą wielu wąskich prostokątów pod krzywą.Wybrana pochodna nie wpływa na ostateczny wynik. Na przykład G z x może różnić się od innej pochodnej o stałą C, ale C znosi się przy odejmowaniu wartości na końcach przedziału.Powszechnym zastosowaniem jest wyznaczenie przemieszczenia netto obiektu za pomocą funkcji prędkości.Przemieszczenie netto dla danego przedziału czasu, zaczynającego od 1 sekundy do 4 sekund, to powierzchnia pod wykresem prędkości i czasu.Aby to obliczyć, najpierw znajdź antypochodną funkcji prędkości — to daje funkcję położenia. Następnie, korzystając z twierdzenia, oceniamy pozycję w końcach przedziału. Odejmij wartość funkcji pozycji po 1 sekundzie od jej wartości po 4 sekundach, aby uzyskać netto przemieszczenie.

podstawowe twierdzenie rachunku całkowego ii

W rachunku r&#xF3;&#x17C;niczkowym i ca&#x142;kowym obliczanie powierzchni pod ci&#x105;g&#x142;&#x105; krzyw&#x105; zosta&#x142;o zasadniczo uproszczone dzi&#x119;ki zastosowaniu Twierdzenia podstawowego rachunku r&#xF3;&#x17C;niczkowego i ca&#x142;kowego, cz&#x119;&#x15B;ci 2. Zamiast polega&#x107; na procesie granicznym sumowania niesko&#x144;czonej liczby niesko&#x144;czenie ma&#x142;ych prostok&#x105;t&#xF3;w, twierdzenie to pozwala na bezpo&#x15B;rednie wyznaczenie ca&#x142;ki za pomoc&#x105; funkcji pierwotnych, upraszczaj&#x105;c tym samym proces ca&#x142;kowania oznaczonego.Twierdzenie podstawowe rachunku r&#xF3;&#x17C;niczkowego i ca&#x142;kowego, cz&#x119;&#x15B;&#x107; 2, stwierdza, &#x17C;e je&#x15B;li funkcja f(x) jest ci&#x105;g&#x142;a na przedziale domkni&#x119;tym [a, b], to ca&#x142;k&#x119; oznaczon&#x105; funkcji f na tym przedziale mo&#x17C;na obliczy&#x107; za pomoc&#x105; dowolnej z jej funkcji pierwotnych F(x). Matematycznie wyra&#x17C;a si&#x119; to nast&#x119;puj&#x105;co:\begin{equation*}\int_a^b f(x)\,dx = F(b) - F(a)\end{equation*}Ten wynik opiera si&#x119; na fakcie, &#x17C;e wszystkie funkcje pierwotne danej funkcji r&#xF3;&#x17C;ni&#x105; si&#x119; jedynie sta&#x142;&#x105;. Poniewa&#x17C; ta sta&#x142;a znosi si&#x119; podczas obliczania r&#xF3;&#x17C;nicy F(b)-F(a), dowolna funkcja pierwotna mo&#x17C;e zosta&#x107; u&#x17C;yta do obliczenia ca&#x142;ki, co znacznie zwi&#x119;ksza elastyczno&#x15B;&#x107; obliczeniow&#x105;.Nale&#x17C;y zauwa&#x17C;y&#x107;, &#x17C;e funkcja f(x) musi by&#x107; ci&#x105;g&#x142;a w ca&#x142;ym przedziale [a, b]. Obecno&#x15B;&#x107; nieci&#x105;g&#x142;o&#x15B;ci lub punkt&#xF3;w, w kt&#xF3;rych funkcja jest nieokre&#x15B;lona w tym przedziale, wyklucza stosowalno&#x15B;&#x107; twierdzenia. W takich przypadkach ca&#x142;ka oznaczona mo&#x17C;e nie by&#x107; dobrze zdefiniowana za pomoc&#x105; tej metody, a wymagane mo&#x17C;e by&#x107; zastosowanie alternatywnych technik lub ostro&#x17C;nego ca&#x142;kowania przedzia&#x142;ami.Twierdzenie to skutecznie przekszta&#x142;ca ca&#x142;kowanie z procesu opartego na sumowaniu niesko&#x144;czonej liczby niesko&#x144;czenie ma&#x142;ych wielko&#x15B;ci w proces obliczania warto&#x15B;ci funkcji w okre&#x15B;lonych punktach ko&#x144;cowych. Stanowi ono podstaw&#x119; rachunku ca&#x142;kowego, umo&#x17C;liwiaj&#x105;c analityczne rozwi&#x105;zania r&#xF3;&#x17C;nych problem&#xF3;w matematycznych, fizycznych i in&#x17C;ynierskich.

Podstawowe Twierdzenie Rachunku Całkowego II

W rachunku r&#xF3;&#x17C;niczkowym i ca&#x142;kowym obliczanie powierzchni pod ci&#x105;g&#x142;&#x105; krzyw&#x105; zosta&#x142;o zasadniczo uproszczone ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Integrals

Całki

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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

area problem

Approximating Area Under a Curve

Approximating area under a curve is a key way to measure regions with curved boundaries. Straight-sided shapes are easy to handle with formulas for rectangles, triangles, and polygons. Curved regions need a different method because standard geometry does not work well on them. The area problem solves this by breaking the region into smaller, simpler shapes. A common approach uses rectangles to estimate the space under a function. Each rectangle gets its height from the function at a chosen...

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Gauss Sums for Riemann Rectangles

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Definite Integrals for Car Motion

Definite integrals help describe a car’s motion over time by linking velocity to displacement. In a velocity-time graph, the definite integral shows how velocity builds up over a chosen time interval. The area under the curve gives the total displacement, so the graph becomes a clear picture of how far the car moves. One key idea is linearity of definite integrals. This property lets you add or subtract velocity functions while keeping their accumulated effect. If a car’s motion is made of...

properties of definite integral ii

Definite Integrals in Motion Analysis

Definite integrals can be used to find displacement from a velocity-time graph. A velocity function v(t) describes how motion changes over time. The definite integral of v(t) from a to b gives the net displacement between those times. It also represents the signed area under the velocity curve between the two points. Two useful properties make definite integrals easier to work with in motion problems. The additivity property says that if a continuous function is integrated over adjacent...

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Definite Integral Rules for Motion

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Definite Integrals: A Step-by-Step Strategy

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Accumulation and Rates in the Fundamental Theorem

The Fundamental Theorem of Calculus connects accumulation and rate of change. In this lesson, the idea is shown with a water tank that is filled by a pump. The pump rate changes over time, so the total water collected must be found by adding up the changing inflow. In water management, this kind of calculation matters for pressure control, operation timing, and system safety. The flow rate is given by a time-dependent function. To find how much water has entered the tank from the start to a...

fundamental theorem of calculus ii

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...

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The Net Change Theorem links a function’s rate of change to its total accumulated change over an interval. In calculus, this means the definite integral of a derivative on [a,b] gives the net change in the original function. The theorem is useful in physics, economics, and engineering because it turns a rate into a total amount. One clear application is a reservoir that receives water from two inflow sources at the same time. The reservoir starts with 260 million cubic meters of water. Each...

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Work Done on a Nonlinear Spring

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substitution rule applied to definite integrals

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Using Substitution to Evaluate Definite Integrals

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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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