This article originated from personal public account: TechFlow, original is not easy, for attention


Today, in the 13th installment of our advanced Mathematics series, we’re going to look at how definite integrals should be computed.


The practical significance of definite integrals


Now, from previous articles, we’ve gotten pretty familiar with the concept of definite integrals and some of their simple properties, and today we’re finally getting down to business, and we’re going to try to evaluate this integral.

Let’s first recall the intuition of a definite integral, which can represent the area of a curve, for example:

If f(x) in the figure above is a function of velocity and the X-axis is time, then f(x) represents the speed of the object at time x. So we add up all the instantaneous distances, and we get the displacement vector of the object over a certain period of time, and that displacement is exactly the area of our curve. When we link the definite integral with the physical displacement, it is easy to conclude that in physics, the displacement of an object is also a one-to-one mapping of time, so it is also a function.

With this conclusion in mind, we can make an assumption that a function s(t) satisfies:


Where a is a constant, we can think of it as a constantThe time at which the displacement beginsS (t) is a function of displacement and time. So the displacement between a and B is equal to.


deduction


When we link the definite integral to the physical displacement, we’re pretty close to solving it.

According to the physical definition, the velocity of the object is actually equal to the rate of change of the position vector with time. Although it is not precise, it is actually a micro component, which can be approximately regarded as the derivative of the displacement function. Of course, this is just intuitive, but we need to express it in rigorous mathematical language.

We assume that the f(x) function is continuous on the interval [a, b], andLet’s try to prove that.

Let’s take something that has a sufficiently small absolute value,, then:


Let’s subtract from thatTo get:


According to our mean value theorem of integrals, we know that alpha existsThat:


Since f(x) is continuous on [a, b], and, so, so, further proofThe derivative of exists, and:


We are now very close to our goal, just one step away. The most important step was claimed by two mathematical giants, Newton and Leibniz. This is one of the most famous cases in mathematics. The background of this story is very complicated, and it is typical of both sides of the argument. There is a famous documentary called “A Calculus History” about this story, if you are interested, you can go to station B and watch it.

To avoid a war, many textbooks call it the Newton-Leibniz formula, named after two people.


Newton-leibniz formula


According to the definition of the antiderivative, we can see from the conclusion above thatIs the functionOne on [a, B]The function. We’re assuming that F of x is also an antiderivative of F of x, so we know thatC is a constant here.

If I set x equal to a, then I get, according to theWe can figure out the definition of, soAnd,, can be obtained by substituting:


If we plug in b, we getThis is Newton’s Leibniz formula.

Let’s review the derivation above, it’s not very difficult, but the substitution is very clever, otherwise even if we can get a conclusion, it’s not very rigorous.


conclusion


With the calculation formula of definite integral, many problems that we could not solve before could be solved, thus laying the foundation of the whole calculus, not only promoted the development of mathematics, but also promoted almost all disciplines of science and engineering. Calculus is used to perform complex calculations in almost every major science and engineering discipline, even in the field of computer science, which seems to be unrelated to mathematics. This is why the university offers this course to all science and engineering students.

But unfortunately, it is often hard to foresee the importance of it when we are learning. However, when we foresee it, it is often many years later, and there is no such environment and time for us to study well.

Today’s article is all of these, if you feel that there is a harvest, please feel free to point attention or forward, your help is very important to me.