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The practical significance of definite integral
Through the previous article, we are basically familiar with the concept of definite integral and some of its simple properties. Today we finally come to the topic, we have to try to calculate this integral.
Let us first recall the intuitive feeling of definite integral, which can represent a curved area, such as:
If we regard f(x) in the above figure as a speed function and the x-axis as time, then f(x) represents the speed of the object's movement at time x. Then we add up all the instantaneous moving distances to get the displacement vector of the object in a certain period of time, and this displacement length is exactly the area of our curved shape. After we link the definite integral to the physical displacement, it is easy to draw a conclusion that in physics, the displacement of an object and time are also a one-to-one mapping relationship, so this is also a function.
With this conclusion, we can make a hypothesis, assuming that a function s(t) satisfies:
s(t)=∫taf(t)dt
where a is a fixed value, we can think of it as the starting moment of displacement, and s(t) is a function of object displacement and time. Therefore, the displacement during the period from a to b is equal to s(b)−s(a)=∫baf(t)dt.
Calculation derivation
When we link definite integral to physical displacement, we are very close to solving it.
According to the physical definition, the moving speed of an object is actually equal to the rate of change of the position vector with time. Although it is not rigorous enough, it is actually a derivative, which can be approximately regarded as the derivative of the displacement function. Of course, this is only an intuitive understanding, we still need to use rigorous mathematical language to express.
We assume that the f(x) function is continuous on the interval [a, b], and Φ(x)=∫xaf(t)dt,(a≤x≤b), we try to prove that Φ′(x)=f( x).
We take an absolute value of Δx that is sufficiently small, such that x+Δx∈(a,b), then:
Φ(x+Δx)=∫x+Δxaf(t)dt
we use it to subtract Φ(x), we get:
ΔΦ=Φ(x+Δx)−Φ(x)=∫x+Δxaf(t)dt−∫xaf(t)dt=∫x+Δxxf(t)dt
According to our integral median theorem, we can get that there is ξ ∈(x,x+Δx), such that:
ΔΦΔΦΔx=f(ξ)Δx=f(ξ)
Since f(x) is continuous on [a, b] and Δx→0, ξ→x, so limΔx→0f(ξ)=f(x), further It proves that the derivative of Φ(x) exists, and:
Φ'(x)=f(x)
is very close to our goal here, only the last step. This most important step has two mathematicians claiming sovereignty over it, one is Newton and the other is Leibniz. This is also a very well-known koan in mathematics. The story behind this is very complicated, and it is a typical bridge between saying that the public is justified. There is a famous documentary called "A History of Complaints and Complaints in Calculus" which tells this story, and interested students can go to station B to watch.
In order to avoid conflict, many textbooks call it the Newton-Leibniz formula, which is named after two people.
Newton-Leibniz formula
According to the definition of the original function, from the above conclusions, we can get that Φ(x) is an original function of the function f(x) on [a, b]. We assume that F(x) is also a primitive function of f(x), so we can know that F(x)−Φ(x)=C, where C is a constant.
Let x = a, then F(a)−Φ(a)=C can be obtained. According to the definition of Φ(x), we can know that Φ(a)=0, so F(a)=C, and Φ(x )=∫xaf(t)dt, we can get:
F(x)−Φ(x)F(x)−∫xaf(t)dt∫xaf(t)dt=C=F(a)=F(x)−F(a)
We can substitute b into it, we can get ∫xaf(x)dx=F(b)−F(a), this formula is the Newton Leibniz formula.
Let's review the above derivation process, it is not difficult, but several substitutions are very clever, otherwise even if we can get a conclusion, it is not rigorous.
to sum up
With the calculation formula of definite integral, many problems that we could not solve before can be solved, thus laying the foundation of the whole calculus, not only promoting the development of mathematics, but also driving almost all disciplines of science and engineering. Almost all major science and engineering disciplines use calculus to perform some complex calculations, even in the computer field that seems not so related to mathematics. This is why the university offers this course for all science and engineering students s reason.
Unfortunately, it is often difficult to foresee its importance when we are studying. However, when we foresee this, it is often many years later. There is no such environment and time for us to study well.