Welcome to the Fundamental Theorem of Calculus! This remarkable theorem establishes a profound connection between two seemingly different concepts: differentiation, which finds rates of change, and integration, which finds areas under curves. The theorem shows that these operations are actually inverse processes, fundamentally linking the concepts of slopes and areas in mathematics.
Part 1 of the Fundamental Theorem states that if F of x is defined as the definite integral of f of t from a constant a to a variable x, then the derivative of F with respect to x equals the original function f of x. Watch as we change x: the area under the curve changes, and the rate of change of this area equals the height of the function at point x.
Part 2 of the Fundamental Theorem provides a practical method for evaluating definite integrals. If F is any antiderivative of f, then the definite integral from a to b equals F of b minus F of a. For example, to find the integral of x squared from 1 to 3, we use the antiderivative x cubed over 3, evaluate it at the endpoints, and subtract to get 26 over 3.
The geometric interpretation reveals the deep connection between slopes and areas. As the point moves along the curve, the shaded area represents the accumulated integral F of x. The red tangent line shows the instantaneous rate of change, which equals the function value at that point. This demonstrates why the derivative of the area function equals the original function, beautifully illustrating the inverse relationship between differentiation and integration.
In summary, the Fundamental Theorem of Calculus establishes the profound connection between differentiation and integration as inverse operations. It provides both theoretical understanding and practical tools for calculation. This theorem has wide applications across physics, economics, and engineering, where we often need to relate rates of change to accumulated quantities. The theorem remains one of the most important and beautiful results in all of mathematics, bridging the gap between slopes and areas in an elegant and powerful way.