Antiderivative

If we know the derivative of a function, the process that we try to find its origin function is the antiderivative.

However, the derivative of a constant is 0, we can not determine which constant is in the origin function, so we can just find a set of functions which all have a derivative like this.

For instance, if f(t)=t is the derivative of F(x)

We need to figure out which function has a derivative equal to t.

By doing the opposite operation of differentiation, it is easy to find out one of the origin function F(t)= ½ t^2

As we know, the derivative of a constant is 0, so we can not determine what the certain original function is. In fact, all the function F (t)=½ t^2 +C (C can be any number) have a derivative equal to f(t)

Integration

Integration can be classified into definite integration and indefinite integration. Indefinite integration exactly has the same meaning of antiderivative. The indefinite integration of a function f, which is also known as the antiderivative of a function f, is a function F such that F′=f, while the definite integral is just a number. The definite integration of a function is equal to the area under the function on a certain domain.

As the picture shows above, the grey part is the indefinite integral of the function f (t).

If we replace variate twith a certain number, it becomes the definite integration of the function f (t).

If we want to find the area of the UBC fountain without using π, how can we do just with the radius r?

We can regard this fountain as a circle.

The question is, without π, just with radius r, how can we get the area of a circle?

First, we divide this circle into eight parts and calculate the area of these four triangles. Then, we can roughly get the area of the circle.

Then, I divide this circle into sixteen parts, only regard the area of these eight triangle as the area of circle, we can find the “extra” area is smaller than last graph.

Then, we make a hypothesis that, if we can divide this circle into 2n parts, regard the area of the triangle as the area of circle, the “extra” area will become smaller and smaller, and our result will be more precise.

The process like this is also like the process of integration.

But the difference is, if we want to integrate a function, we need to cut the area ,which is between x axis and the curve, into many pieces and calculate the area of each pieces and add them together. If the number of pieces is big enough, we will get more precise number of the area between x axis and the curve.