Fundamental Theorem of Calculus, Part 2, Definite Integrals, Basic Proof - Calculus

The Second Fundamental Theorem of Calculus states that for a continuous function f on the interval [a, b], the definite integral from a to b is equal to the difference between the value of any antiderivative F at b and the value of F at a. Its formula is: ∫baf(x)dx = F(b) − F(a), where F'(x) = f(x). This theorem provides a powerful method for evaluating definite integrals by finding an antiderivative and evaluating it at the integral's limits.

💡Here's a breakdown of the theorem:
• The Setup: You start with a function f(x) that is continuous on the interval [a, b].
• The Antiderivative: You find an antiderivative of f(x), which is any function F(x) such that its derivative, F'(x), is equal to f(x).
• The Calculation: The theorem says that the definite integral, which represents the area under the curve of f(x) from a to b, can be calculated by taking the antiderivative F(x), evaluating it at the upper limit b (F(b)), and then subtracting the value of the antiderivative evaluated at the lower limit a (F(a)).

💡Why is it important?
• Simplifies Calculation: It provides a direct method to calculate definite integrals, replacing the more complex process of breaking the area into infinite tiny rectangles (numerical integration).
• Connects Integration and Differentiation: It establishes a fundamental link between the process of integration and differentiation by showing that integrating a function's derivative gives you back the original function (up to a constant difference).
• Applications: It is widely used in various fields to find total changes by integrating a rate of change. For example, if f(x) is a velocity function, the integral of f(x) from a to b gives the total displacement over that time interval.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Fundamental theorem of calculus, part 2

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