Integration

Introduction

Integration is the reverse process of differentiation, finding the original function when given its derivative.
It is used for various applications in mathematics, including finding areas under curves or solving differential equations.

Indefinite integration, also referred to as anti-differentiation, involves finding a function F(x) which has the derivative f(x).
Definite integration involves finding the exact value of the area under the curve from one point to another, represented by limits.

Integration by substitution involves changing the variable in an integral to make it easier to integrate.
Integration by parts, derived from the product rule of differentiation, is used when integrating the product of two functions.
Integration using partial fractions simplifies the function into simpler fractions which are easier to integrate individually.

Numerical methods for integration like Trapezium Rule, Simpson’s Rule, and the Mid-Ordinate Rule are used when a function is too complex to integrate analytically or only numerical data is available.
Trapezium Rule approximates the area under the curve by splitting it into several trapezoids of equal width.
Simpson’s Rule provides a more accurate approximation by using parabolas to approximate each segment of the curve.
Mid-Ordinate Rule uses rectangles under the curve at midpoints to approximate the area.

The accuracy of numerical integration depends on the number and width of the subdivisions; more and narrower subdivisions generally yield more accurate results.
Errors in numerical integration can be reduced by using more precise floating point numbers, increasing the number of intervals, or combining different methods.

Integration is used in many areas of mathematics and physics, such as calculating areas and volumes, solving differential equations, and determining the centroid of a region.
Integration also plays a key role in statistics, where it is used to calculate probability distributions and cumulative distribution functions.