Integrals

Integration is a fundamental concept in calculus that is used to calculate the area under a curve, the length of a curve, or the total value when a quantity is continuously changing.

Definition

The integral of a function f(x) over an interval [a, b] is the signed area between the x-axis and the graph of the function on this interval. It is denoted as ∫ from a to b f(x) dx.

The process of finding integrals is called integration. There are two main types of integrals: definite and indefinite.

• A definite integral has upper and lower limits and returns a number, which represents the signed area.
• An indefinite integral, also known as an antiderivative, has no limits and returns a function (or a family of functions).

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration. It has two parts:

1. The first part shows that integration can be reversed by differentiation.
2. The second part provides an effective method for computing definite integrals without having to calculate the area directly.

Techniques of Integration

There are several techniques of integration, including:

• Basic antiderivatives: For example, the antiderivative of x^n is (1/(n+1)) * x^(n+1).
• Substitution: This is the counterpart to the chain rule for differentiation.
• Integration by parts: This is the counterpart to the product rule for differentiation.
• Partial fractions: This technique is used for integrating rational functions.

Applications

Integrals are used extensively in many areas of mathematics and the physical sciences, including in calculations of area and volume, in solving differential equations, in computing probabilities, and in various ways in statistics, physics, and engineering.