# Applications of Integration

## Integration

Integral calculus, Branch of calculus concerned with the theory and applications of integrals. While differential calculus focuses on rates of change, such as slopes of tangent lines and velocities, integral calculus deals with total size or value, such as lengths, areas, and volumes.

The two branches are connected by the fundamental theorem of calculus, which shows how a definite integral is calculated by using its antiderivative (a function whose rate of change, or derivative, equals the function being integrated). For example, integrating a velocity function yields a distance function, which enables the distance traveled by an object over an interval of time to be calculated.

As a result, much of integral calculus deals with the derivation of formulas for finding antiderivative. The great utility of the subject emanates from its use in solving differential equations.

## What is Integration

Often we know the relationship involving the rate of change of two variables, but we may need to know the direct relationship between the two variables. For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time.

To find this direct relationship, we need to use the process which is opposite to differentiation. This is called integration.

The processes of integration are used in many applications.

The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength.

The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building.

Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface).

## Application of integration

### Application in Engineering

1. An Architect Engineer uses integration in determining the amount of the necessary materials to construct curved shape constructions (e.g. dome over a sports arena) and also to measure the weight of that structure. Calculus is used to improve the architecture not only of buildings but also of important infrastructures such as bridges.
2. In Electrical Engineering, Calculus (Integration) is used to determine the exact length of power cable needed to connect two substations, which are miles away from each other.
3. Space flight engineers frequently use calculus when planning for long missions. To launch an exploratory probe, they must consider the different orbiting velocities of the Earth and the planet the probe is targeted for, as well as other gravitational influences like the sun and the moon.

### Application in Medical Science

1. Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed.

### Application in Physics

1. In Physics, Integration is very much needed. For example, to calculate the Centre of Mass, Centre of Gravity, and Mass Moment of Inertia of a sports utility vehicle.
2. To calculate the velocity and trajectory of an object, predict the position of planets, and understand electromagnetism.

### Application in Statistics

1. Statisticians use calculus to evaluate survey data to help develop business plans for different companies. Because a survey involves many different questions with a range of possible answers, calculus allows a more accurate prediction for the appropriate action.

### Application in Research Analysis

1. An operations research analyst will use calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operating efficiency, increase production, and raise profits.

### Application in Graphics

1. A graphics artist uses calculus to determine how different three-dimensional models will behave when subjected to rapidly changing conditions. It can create a realistic environment for movies or video games.

### Application in Chemistry

1. It is used to determine the rate of a chemical reaction and to determine some necessary information of Radioactive decay reaction.

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