Definite Integral as Area Under a Curve

What is a Definite Integral?

In calculus, the definite integral of a function f(x) over an interval [a, b], denoted as:

∫ab f(x) dx

represents the net signed area between the function's graph and the x-axis, bounded by the vertical lines x = a and x = b.

• "Signed" means that area above the x-axis is considered positive, and area below the x-axis is considered negative.
• If the function f(x) is non-negative on the interval [a, b], then the definite integral directly corresponds to the geometric area under the curve.

The definite integral is fundamentally calculated using the concept of limits, summing the areas of infinitely many infinitesimally thin rectangles under the curve (Riemann sums). The Fundamental Theorem of Calculus provides a powerful way to calculate definite integrals if an antiderivative F(x) (where F'(x) = f(x)) is known:

∫ab f(x) dx = F(b) - F(a)

This simulation helps visualize the area represented by the definite integral for some simple functions.