Find the definition concept of calculus and the formula for finding the area of a curve

The rightmost integral of the x-axis curve minus the leftmost integral is the area enclosed by the curve.

So, s= 0-1 (x-x ; dx= x 2 2-x 3 3 0-1 =1 2-1 3=1 6 (0-1 represents the integral with a definite integral from 0 to 1).

So, the area of the graph enclosed by the curve y=x 2 and y=x = 1 6

Curve area

Mathematically, a curve is defined as: let i be an interval of real numbers, i.e., a non-empty subset of the set of real numbers, then curve c is an image of a continuous function c:i x, where x is a topological space.

Intuitively, the curve can be seen as the trajectory of the movement of the spatial particle. Differential geometry is the study of geometry using calculus. In order to be able to apply the knowledge of calculus, we cannot consider all curves, even continuous curves, because continuity is not necessarily differentiable.

This brings us to the differential curve.

If a plane curve can be expressed as a standard equation.

Then its length is:

Among them, the front Li A and B are the upper and lower limits of X.

If the plane curve can be expressed as a parametric equation.

Then its length is:

What is the definition of the length of a curved orange line in calculus?

In calculus, the length of a curve is defined as the shortest distance between any two points on the curve, that is, the total length of the curve is the sum of the distances passing through each point on the curve.

1. Basic formula: (ax n).'anx^(n-1)(sinx) 'cosx(cosx) 'sinx(e^x) 'e^x(lnx) 'The formula for 1 x integral is their inverse. 2. The basic rules of derivation:

the derivative of the product; the law of derivatives of the quotient; Chained derivation of implicit functions. 3. Basic basic methods.

The Newton-Leibniz formula, also commonly referred to as the basic formula of calculus, reveals the connection between a definite integral and an integrand's original or indefinite integral. It shows that the definite integral of a continuous function over the interval [ a , b ] is equal to the increment of any one of its original functions on the interval [ a , b ].

This provides an efficient and simple calculation method for the given integral, which greatly simplifies the calculation of the definite integral.

Calculus is the branch of mathematics in advanced mathematics that studies the differentiation and integration of functions, as well as related concepts and applications. It is a fundamental subject of mathematics. The content mainly includes limits, differential calculus, integral science and their applications.

Differential calculus consists of the operation of finding derivatives and is a set of theories about the rate of change. It makes it possible to discuss functions, velocities, accelerations, and slopes of curves in a common set of notations. Integralism, including the operation of finding integrals, provides a general set of methods for defining and calculating area, volume, etc.

Newton-Leibniz formula.

Theorem (3): If.

The function baif(x) is a continuum function, then f(x) is a primitive function on the district [a,b].

Note: This DAO formula is known as the Newton-Leibniz formula, and it further reveals the connection between the definite integral and the original function (indefinite integral).

It shows that the definite integral of a continuous function over the interval [a,b] is equal to the increment of any one of its original functions on [a,b]. So it is.

A given integral provides an efficient and easy way to calculate it.

Note: The Newton-Leibniz formula is also commonly referred to as the basic formula of calculus.

1. Basic formula:

ax^n) ' = anx^(n-1)

sinx) ' = cosx

cosx) ' = -sinx

e^x) ' = e^x

lnx) ' = 1/x

The integral formula is their inverse.

2. The basic rules of derivation:

the derivative of the product;

the law of derivatives of the quotient;

Chained derivation of implicit functions.

3. Basic basic methods.

a. Directly insert the above basic formula;

b. Variable substitution method;

c. Partial integral method;

d. Rational fractional integral method;

e. Complex integral method;

f. Complex variable function, remainder integral method;

g. Laplace transform integral method;

h. Various other special integration methods.

Note: The variable substitution method is the main method, which is divided into many types;

The first four methods are the level of general college students;

In addition to the Department of Mathematics, generally speaking, the Department of Physics, the Department of Astronomy, the Department of Electrical Engineering, the Department of Meteorology, the Department of Hydrology, the Department of Oceanography, etc., learn the most, and the above methods are generally learned in the undergraduate program. For general majors, even if you go to graduate school, no.

Will definitely learn. For liberal arts, they generally only understand the concept of integrals, and do not have the ability to disintegrate.