Fundamental Theorem of Calculus – Interactive Visualization of Functions, Integrals, and Derivatives

The Fundamental Theorem of Calculus is one of the most important results in mathematics, because it builds a direct bridge between differentiation and integration, showing that these two operations are essentially inverses of each other.

Part 1: If \( F(x) = \displaystyle \int_{a}^{x} f(t)\,dt \), then \( F'(x) = f(x) \)

Part 2: \( \displaystyle \int_{a}^{b} f(x)\,dx = F(b) - F(a) \), where \( F \) is any antiderivative of \( f \)

This interactive visualization lets you explore and verify both parts of the theorem in real time. As you move point P along the graph of \( f(x) \), observe the following:

The black shaded area under \( f(x) \) from \( 0 \) to \( x \), which represents \( F(x) = \displaystyle \int_0^x f(t)\,dt \) (calculated using Riemann sums)
The blue tangent line to \( F(x) \), whose slope equals \( f(x) \) - a direct demonstration of Part 1 of the theorem

Instructions: Select a function from the dropdown menu and drag the point P to see how the integral changes. The black area under f(x) represents the integral F(x), and the tangent line on F(x) shows that its slope equals f(x), demonstrating the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus
Interactive Visualization of Functions, Integrals, and Derivatives

Function f(x) and Area Under Curve

Integral \( F(x) = \int_0^xf(t) dt \)

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