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In the case of a negative number, it is defined as follows:

1. When f(x) is positive, the definite integral of this function in a certain interval represents the area enclosed by the function above the x-axis.

2. When f(x) is negative in a given interval, the definite integral represents the opposite of the area enclosed by the function below the x-axis, i.e., a negative number.

3. When f(x) is positive and negative in a certain interval, the definite integral represents the value of the area enclosed by the function above the x-axis minus the area enclosed below the x-axis.

Comprehension skills for definite integrals

The definite integral is proposed in relation to the area. We were exposed to the concept of area when we were in elementary school, and it was easy to understand the area of squares, rectangles, triangles, etc. Area can be understood as how much of the flat "space" a flat graphic occupies, just as a sheet contains pixels.

Define the side length of pixels (here as tiny squares) as unit lengths, so that the area formula of the square is understood: the square of the side length, that is, the number of pixels contained in the square, so that the area of a straight and regular figure (rectangle, triangle, trapezoid, etc.) can be understood.

The fundamental theorem of calculus makes it easy to solve the definite integral, which can be solved as the difference between the function values of the original function of the corresponding function in the corresponding interval.

The geometric significance of definite integrals is as follows:Geometric significance: The area enclosed by the integrand and the coordinate axis, the part above the x-axis is positive, and the part below the x-axis is negative, according to the image of cosx in the interval [0,2], the positive and negative areas are equal, so their algebraic sum is equal to 0.

The definite integral has many meanings, it can represent the area of a graph, it can also be associated with physics, the definite integral can be negative, but if you want the area of the graph, you need to use its absolute value.

Considerations for understanding definite integrals:To understand this, we need to pay attention to the relationship between definite and indefinite integrals: if a definite integral exists, it is a concrete value (the area of a curved trapezoid), while an indefinite integral is a functional expression, which only has a mathematical relation (Newton-Leibniz formula) and nothing else.

The positive or negative of the integral depends on the integrand and the interval of the integral, when the integral is used to find the area, the interval of the integral is from large to small and the integrand is positive, so the result is the area.

Integrals are usually divided into two types: definite integrals and indefinite integrals. Intuitively, for a given positive real value function, the definite integral over a real number interval can be understood as the area value (a definite real value) of a curved trapezoid, surrounded by curves, lines, and axes on the coordinate plane.

If the integrals of a function exist and are finite, the function is said to be integrable. Generally speaking, the integrand does not have to have only one variable, and the integral domain can also be a space of different dimensions, or even an abstract space without intuitive geometric meaning. As described above, for a real-valued function f with only one variable x, the integral of f over the closed interval [a,b] is denoted as .

The specific process is as follows:

(1+x^2 )dx

Let x=tant

Original = sect·dtant

sect·tant- tantdsect=sect·tant- tant·tantsectdt=sect·tant- (sec t-1)sectdt=sect·tant- (sec t-sect)dt=sect·tant- sec tdt+ sectdt=sect·tant- sect·dtant + sectdtSo. 2×∫sect·dtant=sect·tant-∫sect·dt

sect·tant-ln|zhuansect+tant|+2c=x√(1+x²)-ln|x+√(1+x²)|2c ie. Original = 1 2x (1+x )-1 2ln|x+√(1+x²)|c

Depending on the integrand and the integral interval, the result will of course be positive and negative. When you use a definite integral to find the area, the definite integral is not written casually, but according to the geometric meaning of the area, the result is naturally positive.

For example, if you know how old you are and how old you are younger than you, in the problem of finding your age, the result of using subtraction is positive, but the result of subtraction itself can also be negative.

The importance of the area integral also lies in the fact that it can locally characterize the existence of the non-tangential limit of the analytic function in a circle at the boundary z=e. To be precise, in addition to the set of zeros, the analytic function within a circle has a sufficient and necessary condition for a non-tangential limit at the boundary z=e.

In addition to the positive and negative aspects of the lead plexus function, it is also related to the integral region, as shown in the figure. The Economic Mathematics team will help you to answer, please match with ** price.

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