{"type":"video","version":"1.0","width":1920,"height":1080,"title":"The Fundamental Theorem of Calculus. - Animated Video By Toon_Hunter - Mango Animate","description":"The Fundamental Theorem of Calculus. animation video uploaded by Toon_Hunter. The Fundamental Theorem of Calculus states that if a function f is continuous on the interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a). In simpler terms, it connects the concepts of integration and differentiation by stating that the integral of a function can be found by evaluating its antiderivative at the limits of integration. This theorem has broad applications in fields such as physics, engineering, and economics.\nTo illustrate this theorem, consider the function f(x) = x^2 and the interval [0, 2]. The antiderivative of f(x) is F(x) = (1\/3)x^3. Applying the Fundamental Theorem of Calculus, we find that the definite integral of f from 0 to 2 is F(2) - F(0) = (1\/3)(2)^3 - (1\/3)(0)^3 = 8\/3. This demonstrates how the theorem can be used to efficiently calculate definite integrals.\nIn summary, the Fundamental Theorem of Calculus is a powerful tool in calculus that establishes a fundamental relationship between integration and differentiation. Understanding and applying this theorem is essential for mastering the concepts of calculus and solving a wide range of mathematical problems. Make your animation and host online for free!","url":"https:\/\/mangoanimate.com\/w\/wb3fpccytfdbm\/the-fundamental-theorem-of-calculus\/wr0ffbmt0fpcc\/","author_name":"Toon_Hunter","author_url":"https:\/\/mangoanimate.com\/homepage\/1edb6803-15e1-6e68-9594-f23c915625cf","provider_name":"Mango Animate","provider_url":"https:\/\/mangoanimate.com","thumbnail_url":"https:\/\/online.mangoanimate.com\/ai\/w\/122362239524559104\/1\/thumb.jpg","thumbnail_width":1920,"thumbnail_height":1080,"html":"