Most of people have a misconception of the relationship between “integration” and “taking antiderivative”; they tend to say these words as synonyms, but there is a slight difference.

In general, “Integral” is a function associate with the original function, which is defined by a limiting process. Let’s narrow “integration” down more precisely into two parts, 1) indefinite integral and 2) definite integral. Indefinite integral means integrating a function without any limit but in definite integral there are upper and lower limits, in the other words we called that the interval of integration.

While an antiderivative just means that to find the functions whom derivative will be our original function. There is a very small difference in between definite integral and antiderivative, but there is clearly a big difference in between indefinite integral and antiderivative. Let’s consider an example:

f(x) = x²

The antiderivative of x² is F(x) = ⅓ x³.

The indefinite integral is ∫ x² dx = F(x) = ⅓ x³ + C, which is almost the antiderivative except c. (where “C” is a constant number.)

On the other hand, we learned about the Fundamental Theorem of Calculus couple weeks ago, where we need to apply the second part of this theorem in to a “definite integral”.

The definite integral, however, is ∫ x² dx from a to b = F(b) – F(a) = ⅓ (b³ – a³).

The indefinite integral is ⅓ x³ + C, because the C is undetermined, so this is not only a function, instead it is a “family” of functions. Deeply thinking an antiderivative of f(x) is just any function whose derivative is f(x). For example, an antiderivative of x^3 is x^4/4, but x^4/4 + 2 is also one of an antiderivative. Despite, when we take an indefinite integral, we are in reality finding “all” the possible antiderivatives at once (as different values of C gives different antiderivatives). So there is subtle difference between them but they clearly are two different things. In additionally, we would say that a definite integral is a number which we could apply the second part of the Fundamental Theorem of Calculus; but an antiderivative is a function which we could apply the first part of the Fundamental Theorem of Calculus.

                                  “Integration”  

The mathematic term, “integral” is often described as the area under a curve of a function. More specifically saying, integral is a multiplication, and “finding the area of a curve” is just one way to describe integral. As its name implies, “multiplication” means breaking down one thing into tons of small pieces and combing all the quantities into a new results. And the “new result” is our Integral. More pieces you have, more accurate result you will get.

Example # 1  We are trying to find an area of a circle by using whole bunch of small rectangle, those rectangles will not completely fill in the circle, there must remain some tiny spaces, but we could get the closest result we want. Less area remaining, more accurate area we find and finally we can reach to the actual area under of the circle.

The technique of integration is not only limited to math but play a vital role in other subjects. For example in physics if we are asked to find the distance of a object that is moving with non-uniform acceleration. If we plot a graph for the motion of the object , velocity on Y-axis and time on x-axis. The curve can be of very complex shape. It would be very difficult to find the distance if we use physics formulas here. But we can simply find the distance by calculating the area under the graph by integration. So as a result Integration can make hard problems easy to solve.