A definite integral is a fundamental concept in calculus that represents the accumulated quantity or total change of a function over a specific interval. Geometrically, it equals the signed area between the function curve and the x-axis from point a to point b.
The rigorous definition of definite integrals comes from Riemann sums. We divide the interval into small subintervals, create rectangles with heights equal to function values, and sum their areas. As we increase the number of rectangles, the approximation becomes more accurate. The definite integral is the limit of this sum as the number of rectangles approaches infinity.
The Fundamental Theorem of Calculus provides an elegant way to calculate definite integrals. Instead of computing complex limits of Riemann sums, we can use antiderivatives. If F is an antiderivative of f, then the definite integral from a to b equals F of b minus F of a. This transforms the geometric problem of finding area into simple algebraic subtraction.
When a function takes both positive and negative values, the definite integral represents signed or net area. Areas above the x-axis contribute positively, while areas below contribute negatively. The integral gives us the total signed accumulation. For example, the integral of sine from 0 to 2π equals zero because the positive and negative areas cancel out exactly.
Definite integrals are powerful tools for solving diverse real-world problems. They calculate areas between curves, volumes of three-dimensional objects, work done by variable forces, and average values of functions. In physics, they determine displacement from velocity and total change from rates. The fundamental concept of accumulation makes integrals essential across mathematics, science, and engineering.