# 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.

## Definite and Indefinite Integration

• 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.

## Techniques of Integration

• 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 Integration

• 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.

## Error and Accuracy in Numerical Integration

• 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.

## Applications of Integration

• 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.