What is the order of the points to exchange quadratic integrals, and what is the order of exchanging

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First, restore the graph of the integration region, and then exchange the integration order to obtain c.

First of all, circle the area represented by xy on the coordinates, and then do the transformation of the xy coordinates.

The answer is c (note the segmentation process).

This type of problem requires a sketch of the integration area in the given integration order, and then determine the integration limit after the change order to complete the solution.

Taking this problem as an example, let's first look at the rightmost integral variable is x, which involves the curves x y and x y, that is, y x and y x 2, so that the first integration is fine. Looking at the last integral, the integral variable is y, and the range of y is from 0 to 1

Thus, according to the variation range of the two curves y x, y x 2 and y is [0,1], the sketch of the integration region can be drawn.

The next step is to determine the point limit after changing the point order.

In order to determine the integration limit of x, the integration region is projected to the x-axis, and the maximum range of the projection region is the integration interval of x. In this problem, the interval [0,1] is obtained by projecting the integration region onto the x-axis, so the lower limit of the integration of x is 0, and the upper limit is 1.

In order to determine the range of change of y, the common practice is to take any point x in the integral interval of x, and make a straight line perpendicular to the x axis through this point, and the straight line passes through the integration region d from bottom to top, where the y value corresponding to the point when the line enters the region (i.e., the intersection point of the straight line and the lower curve) is the lower integral limit of y, and the y value corresponding to the point when the straight line passes through d (the intersection point of the straight line and the upper curve) is the upper integral limit of y. In this problem, at any point x in the integral interval of x [0,1], a straight line perpendicular to the x axis is made, and the straight line enters the region d at the intersection point with y x 2, so the lower integral bound of y is x 2;The above straight line passes out of the region d at the intersection point with y x, so the upper limit of the integration of y is x

The lower limit of the first integral limit (for y integrals) after obtaining the replacement of the integral order is x 2, and the upper limit is x; The lower limit of the last integral (for x integrals) is 0, and the upper limit is 1

The commutative integral order is usually for the reintegration of functions that are more than binary. Take the quadratic integral of a binary function as an example, if dx f(x,y)dy is transformed into dy f(x,y)dx, since the previous score is first to y and then to x, and the latter is just the opposite, this is to exchange the integration order, but the transformed integrand usually needs to be multiplied by a transformed Jacobian determinant, and for definite integrals, it is also necessary to take into account the change of the upper and lower bounds of the integral.

The principle of the order of exchange pointsThe commutative integral, also known as the integral, is one of the three types of integrals contained in the duration equation system obtained by the variational method when the atomic orbitals are linearly combined into molecular orbitals, usually expressed by hab and hba, and it is best to integrate the product as much as possible at one time, that is, the integration region is best to be a connected domain, in this connected domain, there is no need to block the graph, that is, a one-time integration from left to right and then from top to bottom, or a one-time integration from top to bottom and then from left to right.

The commutative integral order is usually aimed at the double integration of functions above the binary level, usually the calculation of the double integral.

Taking the quadratic integral of a binary function as an example, if dx f(x,y)dy is transformed into dy f(x,y)dx, since the previous integral is first integrated to y and then to x, and the subsequent integral is just the opposite, this is the exchange of the integration order, but the transformed integrand usually needs to be multiplied by the Jacobian determinant of a transformation, and for the definite integral, the change of the upper and lower bounds of the integral also needs to be considered.

Details: The key to changing the order of the integrals is to draw the integral region of the double integral according to the integral limit of the given quadratic integral, and then treat it as another type (originally x-type, now as y-type; It was originally y-type, but now it is considered x-type), and then write a quadratic integral of another order.

This problem has two quadratic integrals (integration order: first x and then y), so you need to draw the graph of the two parts of the integration region (both y-shaped regions) separately, then merge the two integration regions into one large region, then treat it as an x-shaped region, and finally write the corresponding quadratic integration (integration order: first y and then x).

The integration area is: Here '<' is directly used to represent 'less than or equal to'

x So, the integration region can be written as:

So, this double integral can be written as:

Upper limit 4 lower limit 1)dy (upper limit y lower limit y) f(x,y)dx

This method is commonly used for general multivariate integrals, and it is not wrong to make a direct way to draw a graph

I'm not a master, so I can't read (o)....

1. Double integral is a principled integral, whether it can be accumulated depends on three aspects:

a. The form of the integrand;

b. Integral area;

c. The order of the points.

2. Iterated integral double integral, which must be written as cumulative integral, can be accumulated.

3. The following example questions give a specific order.

In principle: the first credit, it could be:

a. Function integration to function;

b. Function integration to numbers;

c. Digital integration to numbers;

d. Numeric integration into functions.

The second point must be:

Number integral to number.

4. For the original order of the integration of this question, please refer to the division method of the first ** below;

For the division method after changing the order of points, please refer to the diagram on the second sheet**.

Each one** can be clicked to enlarge;

If you have any questions, just ask, and answer them.

1. This question integrates y first, which is not the original integral, but after exchanging the order of the integrals, the integrals can be easily accumulated.

This is a very typical question:

He illustrates that the double integral, which has its own unique two properties:

1. Whether the points can be scored depends on the integration range;

Second, it is more important to look at the order of points.

2. The specific answers are as follows, if you have any questions, please feel free to ask, answer questions, and explain if you have questions.