What are the problems and specific steps of the definite integral element method?

The element method is an analytical method that uses the theory of definite integrals to analyze and solve some problems in geometry and physics, and needs to express a quantity into a definite integral.

Steps. Generally speaking, if the quantity u sought in a practical problem meets the following conditions:

1) u is a variable x interval [a,b];

2) U is additive to the interval [a,b], that is, if the interval [a,b] is divided into many small intervals, then u is correspondingly divided into many partial quantities, and u is equal to the sum of all partial quantities;

3) The approximate value of the partial quantity can be expressed as , then the definite integral can be considered to express this quantity u. Usually the step to write the integral expression of this quantity u is:

1) According to the specific situation of the problem, select a variable such as x as the integral variable, and determine its variation interval [a, b];

2) Suppose that the interval [a,b] is divided into n small intervals, and any one of the intervals is selected and denoted as [x,x+dx], and the approximate value of the partial quantity u corresponding to this cell is obtained. If u can be approximated as the product of the value f(x) of a continuous function at x on [a,b] and the definite integral dx, then the element called the quantity u is denoted as du, i.e.

3) Taking the element of the quantity u as the product expression, the definite integral is made on the interval [a, b] to obtain.

This is the integral expression of the quantity u sought.

[t,t+dt] in the elemental method is actually [t,t+δt] (because the differentiation of the independent variable is equal to the amount of change in the independent variable), but it is written as dt to meet the formal needs of differential equations or definite integrals. The meaning of DM is fundamentally different from DT, it is the differentiation of functions. DV is a microelement in the derivation of the formula using the element method to find the volume, which seems to be the same as DT in name, but in fact, when calculating the volume, the specific expression of DV is required first, that is, when the volume is actually calculated, DV is the differentiation of the function.

So, differentiation plays different roles at different stages, and that's where it differs from dt.

The principle of the elemental method of definite integrals: the principle of the microelement method and the principle of infinite approximation.

1. Detailed explanation:

The micro-element method refers to the method of starting from the analysis of a very small part of a thing (micro-element) when dealing with a problem, and achieving the overall purpose of solving the thing. It is commonly used in solving physics problems, and the idea is to "break the whole into parts", first analyze the "micro-elements", and then analyze the whole through the "micro-elements".

The micro-element method is a common method in analyzing and solving physical problems, and it is also a thinking method from parts to wholes. With this method, some complex physical processes can be quickly solved with the physical laws that we are familiar with, and the problems sought can be simplified.

When using the micro-element method to deal with a problem, it needs to be broken down into many tiny "meta-processes", and each "meta-process" follows the same law, so that we only need to analyze these "meta-processes", and then debate the "meta-processes" to process the necessary mathematical methods or physical ideas to solve the problem. The use of this method strengthens our rethinking of known laws, which leads to the effect of consolidating knowledge, deepening knowledge and improving ability.

2. The elemental method of definite integrals.

The method and steps of using definite integrals to find the virtual destruction product of curved trapezoidal surfaces are almost the same:

Let f(x) be continuous over the interval [a,b] and f(x) 0, and find the area a. of the curved trapezoidal on [a,b] with the curve y=f(x) as the edgeThe idea of expressing this area a as a definite integral width=11,height=14, dpi=110f(x)dx is to "divide, approximate, sum, and limit", and the specific steps are:

1) Segmentation: Divide [a,b] into n intervals, and accordingly divide the curved trapezoidal into n small curved trapezoids, and its area is denoted as δai(i=1,2,...,n).

2) Approximation: Calculate the approximate value of δai for the area over each interval.

ai≈f(ξi)δxi (xi-1≤ξi≤xn)

3) Summation: Obtain an approximate value of area a.

width=111,height=29,dpi=110

4) Take the limit: the exact value of the area a.

The elemental method of definite integral is an analytical method that applies the theory of definite integral to analyze and solve some problems in geometry and physics, and it is necessary to express a quantitative brother into an analytical method of definite integral.

Application Procedure: Generally, if the quantity u in a practical problem meets the following conditions:

1) u is a variable x interval [a,b]; Keitao.

2) U is additive to the interval [a,b], that is, if the interval [a,b] is divided into many small intervals, then u is correspondingly divided into many partial quantities, and u is equal to the sum of all partial quantities;

3. According to the specific situation of the problem, select a variable such as x as the integral variable, and determine its variation interval [a, b].

The element method of definite integral is an analytical method that needs to express a quantity into a definite integral when applying the theory of definite integral to analyze and solve some problems in geometry and physics.

Definite integrals provide an effective way to solve practical problems. If the integrand and integral intervals are known, then the process of solving the definite integral can be summarized as follows: Dividing, Approximating, Summation, and Limiting.

The integrand and integral intervals can be obtained according to practical problems, and an efficient way to find the expression of the integrand is provided by the element method. The Elemental Method, also known as the Element Method, is widely used in the fields of geometry, mechanics, electromagnetism, and medicine, and has gradually penetrated into theoretical mechanics, physics, and other professions.

Definite integral element method: "element method" is a scientific thinking method that divides the research object into infinitely small parts with high and infinitesimal disadvantages, takes out a representative very small part for analysis and processing, and then considers it comprehensively from the part to the whole, which fully embodies the idea of integration in this method.

The element method of definite integral is an analytical method that needs to express a quantity into a definite integral when applying the theory of definite integral to analyze and solve some problems in geometry and physics.

Geometric applications, for example: the calculation of the area of a plane figure, the volume of a rotating body, the rotation around the x-axis, and the point of the integral sheet is a cylinder. The application of the physical butan, for example:

The displacement of variable speed linear motion, let the object do variable speed linear motion to find the displacement of the object in the time interval.

<>f，f'The component forces in the horizontal direction cancel each other out, leaving only the component forces in the vertical direction.

The gravitational force of the X segment to A δf = K 1 δx (A +X ) the gravitational force of an infinite line L to A = 2 (0, + K (A +X ) A (A +X ) Dx

Let x=atan, repentance [0, 2].

2 (0, 2) k a cos d(atan )2k a quietly high (0, 2) cos d 2k a sin (0, 2).

2k/a

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