The essence of double integral: representing the volume of a curved top cylinder.

The essence of triple integral: representing the mass of a solid.

2, the overview of the two is different:

Overview of double integral: double integral is the integral of binary function in space, similar to definite integral, and it is the limit of some form of sum. The essence is to find the volume of a curved top cylinder. Multiple integrals have a wide range of applications, which can be used to calculate the area of curved surface, the center of gravity of plane thin plate and so on. The double integral of plane region can be extended to the integral on the (directed) surface in high-dimensional space, which is called surface integral.

Overview of triple integration: Let the ternary function f(x, y, z) have a first-order continuous partial derivative in the region ω, and divide ω into n small regions at will, and the diameter of each small region is denoted as r? (I = 1, 2, ..., n), the volume is δ δ? ，||T||=max{r？ };

Take a little f(ξ? ,η? ,ζ? ), and σ f (ξ? ,η? ,ζ)Δδ? If the limit of the summation formula exists and is unique when ||||→ 0, it is called the triple integral of the function f(x, y, z) on the area ω, and it is denoted as ∫∫∫f(x, y, z)dV, where dv = dxdydz.

3. Their mathematical meanings are different:

Mathematical significance of double integral: In the spatial rectangular coordinate system, double integral is the algebraic sum of cylinder volumes of all parts in the region, which is positive above xoy plane and negative below xoy plane. The volume formula of the curved cylinder surrounded by the curved surface of some special integrand f(x, y) and the bottom surface of d is known, which can be calculated by the geometric meaning of double integral.

Mathematical significance of triple integral: If a closed region G is divided into finite sub-closed regions by finite surfaces, then the triple integral on G is equal to the sum of triple integrals on each closed region.

4. Different uses

Neither double integral nor triple integral can be used to calculate volume. Double integral can be used to calculate volume, and triple integral cannot be used to calculate volume.