Integration is one of the most important concepts in calculus. It is the reverse operation of differentiation. While differentiation tells us the rate of change at any point, integration helps us accumulate or sum up quantities over an interval. One of the most common applications is finding the area under a curve, as shown in this example.
Indefinite integration is about finding antiderivatives. An antiderivative of a function f(x) is another function F(x) whose derivative equals f(x). For example, if we know that the derivative of x squared is 2x, then the integral of 2x is x squared plus a constant C. Let's look at an example: the integral of x cubed equals x to the fourth power divided by 4, plus C. We can verify this by taking the derivative and getting back x cubed.
Definite integration calculates the exact numerical value of an integral over a specific interval from a to b. Unlike indefinite integration, the result is a number, not a function. The definite integral represents the area under the curve between the limits of integration. For example, the definite integral of one-half x plus one-half from 1 to 3 equals 3, which is the shaded area under the line.
Riemann sums provide a way to understand definite integration geometrically. We approximate the area under a curve by dividing the region into rectangles. Starting with just a few rectangles, we can see the approximation is rough. But as we increase the number of rectangles, the approximation becomes more and more accurate. In the limit as the number of rectangles approaches infinity, we get the exact value of the definite integral.
Integration has numerous practical applications across mathematics, physics, and engineering. We can use integration to find areas between curves, calculate volumes of three-dimensional objects formed by rotating curves, and solve physics problems. For example, we can find displacement by integrating velocity over time, or calculate the work done by a variable force. Integration is also essential for finding centers of mass and solving differential equations that model real-world phenomena.