Integration is one of the fundamental operations in calculus. It serves as the reverse of differentiation, allowing us to find the original function when we know its derivative. Geometrically, integration calculates the area under a curve. The integral symbol represents this process of finding areas and antiderivatives. We can approximate this area using rectangles, and as we use more rectangles, we get closer to the exact area.
Riemann sums provide the mathematical foundation for integration. We divide the area under a curve into rectangles of equal width. The sum of these rectangular areas approximates the total area. As we increase the number of rectangles, making them thinner, our approximation becomes more accurate. In the limit as the number of rectangles approaches infinity, we get the exact value of the definite integral.
The Fundamental Theorem of Calculus establishes the crucial connection between derivatives and integrals. It states that if F is an antiderivative of f, then the definite integral from a to b equals F of b minus F of a. This theorem allows us to evaluate definite integrals without using Riemann sums. For example, to find the integral of x squared from 1 to 3, we use the antiderivative x cubed over 3, evaluate it at the bounds, and subtract to get 26 over 3.
There are several key techniques for finding antiderivatives. The power rule states that the integral of x to the n equals x to the n plus 1, divided by n plus 1, plus a constant. Substitution helps with composite functions by replacing complex expressions with simpler variables. Integration by parts uses the product rule in reverse. We also have basic formulas for common functions like exponentials, trigonometric functions, and logarithms. These techniques form the foundation for solving most integration problems.