Integration is one of the fundamental concepts in calculus. It is the reverse process of differentiation. While differentiation finds the rate of change of a function, integration finds the original function from its rate of change. The integral sign looks like an elongated S, representing the sum of infinitely small parts.
Indefinite integration finds the antiderivative of a function. The result includes a constant of integration, C, because when we differentiate, constants disappear. This means there are infinitely many functions that have the same derivative, differing only by a constant. Watch how changing C shifts the parabola up and down.
Definite integration calculates the exact area under a curve between two points. We can approximate this area using Riemann sums, which divide the region into rectangles. As we use more and thinner rectangles, the sum of their areas approaches the exact area under the curve. This is the fundamental idea behind definite integration.
Integration follows several basic rules. The power rule states that the integral of x to the n is x to the n plus one divided by n plus one, plus C. The constant rule shows that the integral of a constant k is k times x plus C. The sum rule allows us to integrate each term separately. Let's see an example: integrating three x squared plus two x plus one gives us x cubed plus x squared plus x plus C.
To summarize what we have learned about integration: Integration is the fundamental reverse process of differentiation. Indefinite integration finds families of antiderivatives, while definite integration calculates exact areas under curves. These concepts form the foundation of calculus and have countless applications in mathematics, physics, and engineering.