Integration is one of the fundamental concepts in calculus. It is essentially the reverse process of differentiation. While differentiation finds the rate of change, integration finds the total accumulation of a quantity. The most common application is finding the area under a curve, as shown by the shaded region in this graph.
There are two main types of integrals in calculus. The indefinite integral represents the family of all antiderivatives of a function and includes an arbitrary constant C. The definite integral, on the other hand, calculates the net area between the curve and the x-axis over a specific interval from a to b, giving us a single numerical value.
Integration can be understood through Riemann sums. We approximate the area under a curve by dividing it into rectangles and summing their areas. As we increase the number of rectangles, making them thinner and thinner, the approximation becomes more accurate. In the limit, as the number of rectangles approaches infinity, this sum gives us the exact value of the definite integral.
The Fundamental Theorem of Calculus is the bridge that connects differentiation and integration. It states that if F of x is an antiderivative of f of x, then the definite integral from a to b of f of x dx equals F of b minus F of a. This powerful theorem allows us to evaluate definite integrals by finding antiderivatives, rather than computing limits of Riemann sums.
Integration has numerous practical applications across many fields. In geometry, we use it to find areas between curves and volumes of solids of revolution. In physics, integration helps calculate work done by variable forces, displacement from velocity, and centers of mass. In economics, it's used for consumer and producer surplus. In probability and statistics, integration is essential for continuous probability distributions. These applications make integration one of the most powerful tools in mathematics and science.