First, the main points:
1. Overview of definite integral: As an integral, definite integral is the limit of the integral sum of function F (x) in the interval [a, b].
2. Overview of double integral: Double integral is the integral of binary function in space, which is similar to definite integral and is the limit of specific sum. Its essence is to find the volume of a cylinder with a curved top. Multiple integrals are widely used to calculate the surface area and center of gravity of plane slices.
3. Overview of triple integration: The first-order continuous partial derivative of ω in the region of ternary function f (x, y, z), ω is arbitrarily divided into n small regions, and the diameter of each small region is r? Remember (I = 1, 2, ..., n).
Volume record δ δ? | | T | | = Max？ {r}, in each small f region (factor? ,η? ,ζ? ), as a permanent σ f (factor? ,η? ,ζ? )Δδ? If the limit of | | T | |-> 0 in the formula exists and is unique (i.e., it has nothing to do with choosing the splitting point ω);
It is called the triple integral of the limit function f (x, y, z). Remember ∫∫∫f (x, y, z) dV, where dV = dxdydz.
Second, the geometric significance:
1, the geometric meaning of definite integral: representing the area of a plane figure.
2. The geometric meaning of the double integral: it represents the volume of the cylinder at the top of the curved surface.
3. The geometric meaning of triple integral: it represents the mass of solid.
Third, the preventive measures are different:
1. Note on definite integral: For a function, there can be indefinite integral, but there can be no definite integral: there can be definite integral, but there can be no indefinite integral. For continuous functions, there must be definite integral and indefinite integral: if there are only a finite number of discontinuous points, definite integral exists. If there is a jump breakpoint, then the function must not exist, that is, the indefinite integral must not exist.
2. Note on the double integral: The double integral on the plane area can be extended to the integral on the high-dimensional space (directed surface), which is called surface integral.
3. Note for the third integration: When the integration function is 1, the density is evenly distributed, which is 1, and the mass is equal to its volume value. When the integral function is not 1, the density distribution is uneven.
Definite integral, double integral and triple integral are important contents in higher mathematics, in which definite integral is the basis of learning double integral and triple integral.
References:
Baidu Encyclopedia-Double Integral
References:
Baidu Encyclopedia-Triple Integral