How to calculate the triple integral problem, how to solve the triple integral?

First of all, you need to understand the basic shape of the integration region. That is, you know some of the common shapes of spatial surfaces that make up the integration region.

In this problem, Z=X 2+2Y 2, it is a rotating parabola Z=X 2+Y 2 with an opening on the Z axis, and the Y scale is magnified, so the shape is basically unchanged, and it crosses the coordinate origin.

z=2-x 2 is a parabolic cylinder with an opening pointing downwards and passing (0,0,2) points.

Then the upper and lower limits of the z-integral are determined, the lower limit is the rotating parabola z=z=x 2+2y 2, and the upper limit is the parabolic cylinder z=2-x 2.

There are generally two methods for solving triple integrals, the projection method and the cross-section method, both of which use the microelement analysis method to calculate the mass of a spatial inhomogeneous geometry.

1. Projection method solution steps. Projection, as the name suggests, is to find the projection of a given geometry first. The specific steps can be seen in the following figure:

2. The solution steps of the cross-section method. When calculating some practical problems, sometimes the stool uses the projection method to calculate the triple integral, which will be very computational, and even the integration will be difficult. In this case, the cross-section method greatly simplifies the calculation process. The specific steps are as follows:

3. Explanation of the cross-section method. If the integrative Wang function in the triple integral is not related to x,y, it is easier to calculate the cross-sectional area obtained by using a plane parallel to the xoy coordinate plane to intercept the closed region.

4. Further explanation of the projection method. The integrand function is related to x, y, and z, and can generally be calculated by the projection method.

There are two columnar approaches to this type of question:

1. In the cross-section method, 0 z 1 is determined first, and then the integral area is intercepted with a cross-section perpendicular to the z-axis to obtain the red cross-section in the figure below, and the entire integration area is stacked by layers of cross-sections. This process can also be referred to as the first-two-after-one method.

2. Projection method, first determine the maximum projection surface x 2+y 2=1, and then use a line of vertical projection surface to pass through the integration area to obtain some columns of curved top cylinders, and the entire integration area is piled up by one cylinder. This process can also be referred to as the first-to-the-second method.

Y=X, X=1 and Z=0 represent the boundaries of the three planes of Bai, respectively, so it can be seen that Zhi is in the X-Y plane, and DAO can determine the range of X and Y.

special x (0,1), y (0,x). You can see this by drawing a diagram. In fact, you can come up with a figure in your mind without drawing.

That's how I'm sure. Also, you don't have to draw the entire 3D diagram at once, which makes it difficult to determine the range. It's more intuitive to divide into a certain plane.

In this case, it would be helpful to draw only the x-y plane, and the range of z can be determined by z=xy, z=0. i.e. z (0,xy).

This is true for the use of symmetry.

Intuitively, it's easy to take it apart and simplify it.

The solution is as follows. Using symmetry, the original formula = 8 (0,2)dx (0,2)dy (0,2)(x+y+z))dz.

Whereas, (0,2)(x+y+z))dz=2(x+y+1). Original = 16 (0,2)DX (0,2)(X+Y+1)dy=32 (0,2)(X+2)DX=...=192。

In the problem, there is z greater than or equal to the x square plus y square under the root number. Therefore, the lower bound of z in the column coordinate solution should be p.

This error is hard to find, but it's also easy to make mistakes, so I'll underline the part. When opening the root number there, you must ensure that it is not negative.