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Usage of integrals and ratios in real life

As we know, speed is a measure of how quickly an object or body moves from one place to another. Mathematically, it is equal to the distance travelled by the time. We can represent the relationship between time, speed and distance when exactly one of them is constant.

  • When the distance travelled is constant, the speed is inversely proportional to the time taken, i.e. if time increases, speed will be reduced and vice versa. 
  • When the time taken to cover the distance is constant, then the distance is directly proportional to the speed. 
  • Similarly, when the speed is constant, the distance will be directly proportional to the time taken.

When the distance travelled by two persons and the time taken by them is given, then we can conclude the fastest of them by comparison of ratios of distance to time. Also, the concept of ratios is a very important tool in solving various aptitude based problems.

We often use the relationship between distance, velocity and acceleration in most of the problems. Where distance is the measurement of how far the objects or things from a given point, acceleration is the rate of change of velocity with given time. The concept of indefinite integrals is used in the problems involving distance, velocity acceleration when each of them is a function of time.

It can be said that, the derivative of a distance function represents instantaneous velocity and that the derivative of the velocity function represents instantaneous acceleration at a particular time. By considering the relationship between the derivative and the indefinite integral as inverse operations, remember that the indefinite integral of the acceleration function represents the velocity function and similarly, the distance function is equal to the indefinite integral of the velocity.

There is another case, where the time interval is given for the function of time. In this case definite integral should be used instead of indefinite integral. Also, in definite integral calculation one must avoid adding constant of integration, and it will give the definite value due to the limits on integration.

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