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

The second part of the Fundamental Theorem of Calculus connects antiderivatives with definite integrals. It finds the exact total accumulation of a function's rate over a specific interval.If a function is continuous on a closed interval, the definite integral equals the antiderivative's value at the upper limit minus its value at the lower limit.This avoids Riemann sum approximations, which estimate the area using many narrow rectangles under the curve.The antiderivative chosen does not affect the final result. For instance, G of x may differ from another antiderivative by a constant C, but C cancels when subtracting values at the interval&#8217;s endpoints.A common application is finding an object&#8217;s net displacement using its velocity function.The net displacement for a given time interval, starting from 1 second to 4 seconds, is the area under the velocity-time graph.To calculate this, first find the antiderivative of the velocity function&#8212;this gives the position function. Then, using the theorem, evaluate the position at the interval&#8217;s endpoints. Subtract the value of the position function at 1 second from its value at 4 seconds to get the net displacement.

fundamental theorem of calculus ii

In calculus, the computation of the area under a continuous curve has been fundamentally simplified by applying the Fundamental Theorem of Calculus, Part 2. Rather than relying on the limiting process of summing infinitely many infinitesimal rectangles, this theorem permits direct evaluation using antiderivatives, thereby streamlining the process of definite integration.The Fundamental Theorem of Calculus, Part 2, states that if a function f(x) is continuous on a closed interval [a, b], then the definite integral of f over that interval can be computed using any of its antiderivatives F(x). Mathematically, this is expressed as:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:35.09%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19982/19982_Resized_equation_editor_19982.svg" alt="Fundamental theorem of calculus; formula, integral &#8747;ab f(x)dx=F(b)-F(a), mathematical concept." data-altupdatedat="1775223722000" data-alt="19982_Resized_equation.svg" /></figure>This result depends on the fact that all antiderivatives of a function differ only by a constant. Since this constant cancels out when evaluating the difference F(b) - F(a), any antiderivative can calculate the integral, greatly enhancing computational flexibility.It is crucial to note that the function f(x) must be continuous throughout the interval [a, b]. The presence of discontinuities or points where the function is undefined within the interval nullifies the theorem&#8217;s applicability. In such cases, the definite integral may not be well-defined using this method, and alternative techniques or careful piecewise integration must be employed.This theorem effectively transforms integration from a process based on summing infinitesimal quantities to evaluating function values at specific endpoints. It serves as a cornerstone of integral calculus, enabling analytical solutions to various problems in mathematics, physics, and engineering

Fundamental Theorem of Calculus II

In calculus, the computation of the area under a continuous curve has been fundamentally simplified by applying the Fundamental Theorem of Calculus, Part ...

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Calculus

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Integrals

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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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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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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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