Integrals.

An integral is a mathematical concept that represents the area under a curve in a graph. It is a fundamental concept in calculus and is used to calculate quantities such as area, volume, and displacement. Integrals can be thought of as the reverse operation of differentiation, and they are denoted by the symbol ∫. There are two main types of integrals: definite integrals, which have specific limits of integration, and indefinite integrals, which do not have specific limits and result in a function with an arbitrary constant of integration. Integrals are used in a wide range of fields, including physics, engineering, economics, and statistics.

Definition:
An integral is a mathematical operation that finds the area under the curve of a function over a given interval.

Notation:
∫[a,b] f(x) dx

where:
- f(x) is the function to be integrated
- a is the lower limit of integration
- b is the upper limit of integration

Types of Integrals:
- Indefinite Integral: Finds the antiderivative of a function.
- Definite Integral: Finds the area under the curve between two specific points.

Fundamental Theorem of Calculus:
The Fundamental Theorem of Calculus states that the definite integral of a function over an interval is equal to the net change in the function's antiderivative over that interval.

Steps to Solve Integrals:
1. Find the antiderivative of f(x).
2. Evaluate the antiderivative at the upper limit of integration.
3. Evaluate the antiderivative at the lower limit of integration.
4. Subtract the result of step 3 from the result of step 2.

Common Integral Forms:
- ∫x^n dx = (x^(n+1))/(n+1) + C (where C is the constant of integration)
- ∫sinx dx = -cosx + C
- ∫cosx dx = sinx + C
- ∫ex dx = ex + C
- ∫(1/x) dx = ln|x| + C

Applications of Integrals:
- Finding areas and volumes of regions
- Calculating work done by a force
- Finding probabilities in statistics
- Solving differential equations
- Modeling real-world phenomena