Definite or Indefinite Integrals

Introduction

Calculus is a branch of mathematics developed from algebra and geometry built on two major complementary ideas. The first is differentiation and the second is integration. Here we deals with integration which is about a vast generation of area. Though it is motivated by area, it includes concepts such as volume and even distance. There are two types of integrals. Definite integrals and indefinite integrals.

Definite integrals

It is the most basic form of integral. We can explain it through an example. The change in position of a car travelling on a highway overtime. The rate of change of the car overtime in this case, its velocity is increasing. If one wants to add the rate of change of car's velocity, one takes the integral of the rate of change of the car's position overtime. This is the process of integration. Another example provides for the rate of change of the car with respect to its position on the road to vary. That is, the first two seconds, let us allow the car's velocity to be increasing. Then, the next four seconds allow the car's velocity to be decreasing. Then allow it to increase again for the next two seconds. Using integration on the velocity of the car over 8 seconds will yield the car's net change of position compared to its initial position at the time one started to measure the different rates of change of the car's position with respect to the road.

Indefinite integrals

It is another form of integral. If possible, a general formula that gives one a family of curves whose slope at any time is provided by the answer of the integral also called the antiderivative. For example: If you find the integral of 1, one is trying to find the equation of a function whose slope at every point is 1. The answer is the line y = x + k where k is a constant because the slope of y = x + k is always 1. So the family of curves y = x + k all have a slope of 1 everywhere on the line.