Integration is a fundamental concept in calculus. It is the reverse process of differentiation, also known as antidifferentiation. When we integrate a function f of x, we find another function F of x whose derivative equals the original function.
The geometric interpretation of integration is finding the area under a curve. When we compute the definite integral from a to b, we get the exact area between the function and the x-axis over that interval. This shaded region represents the value of the integral.
There are two main types of integration. Indefinite integration finds the antiderivative of a function, which includes an arbitrary constant C. Definite integration calculates a specific numerical value representing the area under the curve between two limits.
Integration can be understood through Riemann sums. We divide the area under the curve into rectangles. As we increase the number of rectangles, our approximation becomes more accurate. In the limit, as the number of rectangles approaches infinity, we get the exact value of the integral.
To summarize what we have learned: Integration is the reverse process of differentiation. It helps us find areas under curves and antiderivatives. There are two main types - indefinite integration which includes a constant, and definite integration which gives a numerical value. Integration is a fundamental tool in calculus with applications across mathematics, physics, and engineering.