An integral is a fundamental concept in calculus with two main interpretations. First, as an antiderivative or indefinite integral, it represents the reverse process of differentiation. Second, as a definite integral, it calculates the signed area between a function and the x-axis over a specific interval, as shown by the yellow shaded region.
The indefinite integral, also called the antiderivative, finds a function F of x whose derivative equals the given function f of x. The general form is the integral of f of x dx equals F of x plus C, where F prime of x equals f of x and C is an arbitrary constant. For example, the integral of 2x dx equals x squared plus C, because the derivative of x squared plus C is 2x. The graph shows the derivative function in red and the family of antiderivatives in blue, differing only by constants.
The definite integral calculates the signed area between a curve and the x-axis over a specific interval from a to b. We can approximate this area using Riemann sums, which divide the region into rectangles. As we increase the number of rectangles and make them infinitely thin, the sum approaches the exact area under the curve. This is the fundamental idea behind the definite integral.
The Fundamental Theorem of Calculus is the bridge that connects differentiation and integration. Part 1 states that the derivative of an integral is the original function. Part 2 shows that a definite integral can be evaluated using antiderivatives: the integral from a to b of f of x dx equals F of b minus F of a, where F is an antiderivative of f. This remarkable theorem allows us to calculate exact areas using antiderivatives instead of limits of Riemann sums.
To summarize what we have learned about integrals: They have two main interpretations as antiderivatives and area calculations. Indefinite integrals reverse the differentiation process to find families of functions. Definite integrals calculate the signed area between curves and the x-axis. The Fundamental Theorem of Calculus elegantly connects these concepts, making integrals an essential tool in mathematics, physics, and engineering.