The Difference and Connection between Antiderivative and Integral

The antiderivative is the inverse process of the derivative.The integral is to calculate the area closed by the graph of a function and x(or y) axis. However, Newton-Leibniz connect antiderivative and integration by giving a formula:∫^a_bf (x) dx = F(b)−F(a). In some cases, the integral of function f(x) on closed interval [a,b] can be calculated from the difference of its bounds’ antiderivative[(F(a)-F(b)]. When we do the calculation of antiderivative, we come up with a formula with an inconstant instead a definite value, which is the outcome of integral. A continues function must have its antiderivative. A function with first discontinuous points does not have its antiderivative.

Antiderivative and integral are two different things: if on interval[a,b], F’(x)=f(x), we can say F(x) is one of antiderivatives of f(x) on this interval. Based on this case, if F(x) is one of antiderivatives of f(x), F(x)+C(C is a random constant) is antiderivative of f(x). Then if f(x) is able to be integrated on [a,b], it may not have antiderivative. If f(x) is continuous on [a,b], f(x) constantly has its antiderivative G(x) and G(x) is its upper limit function: F(x)=∫^x_af(t)dt+C

In fact, the antiderivative equals indefinite integral. A continuous function can have infinite antiderivatives, but can have only 1 definite value of integral(interval must be given).

The Motivation of Integration

    When you drive to work every day, have you ever think about how does your driving speed affect the time you spend on the road? You may think: well, it’s super simple the question.My house and the office building cannot walk so the distance will never change. When I drive faster, the time it takes me on the road is shorter than that when I drive slower. Like the lower-upper level relation between you and your boss, the distance you daily drive is the lower level of the exact speed of your car. That means the speed determine how the distance work as your boss determine how you work.

    Once you are given a task from your boss, generally you should break the task into many small parts. It’s easy to manipulate most of the things step by step. Let us roughly and simply break the task into three steps: the first step to begin, the second step to work for the goal and the third step to close. As long one of these three steps has been done in order, 20%, 80%, 100% of the process are completed separately. The complement of each step needs time to be pushed forward. But if we add up all these parts, the time goes on and the process is accumulating, then the whole task will be done which means you can earn your salary even can hear a raise from your boss. Actually, we can see the speed and distance work in a similar way: the engineer pushes your car in three constant speed. Going with different driving speed and flowing time, the street runs behind you. Finally, you either catch up on time or come late and lose the attendance bonus (sad). Whatever, the car has finished the aimed mileage, its task.

The sum up of stepped speed shows the distance. The boss determines how much work you should do. The upper level instructs lower level. The lower level in some fields is not easy to manipulate. At that time, we ought to focus on the upper level, because it provides a clearer overview of what is going on now in the lower level.