Triple Integral

Definition. Consider a function f(x, y, z) defined at every point of the 3-dimensional finite region V. Divide V into n elementary volumes δV1, δV2, ………. δVn. Let (xr, yr, zr) be any point within the rth sub-division δVr .

Consider the sum ∑_(r=1)∞ f(x_r, y_r,z_r,) δV_r
The limit of this sum, if it exists, as n→∞ and δVr → 0 is called the triple integral of f(x, y, z) over the region V. It is denoted by
        ∫ ∫ ∫ f(x, y, z) dV.

Evaluation of Triple Integrals
For purpose of evaluation, the triple integral ∫ ∫ ∫ f(x, y, z) dV is expressed as a repeated integral
        ∫ ∫ ∫_x1^x2 ∫ ∫ ∫_y1^y∫ ∫ ∫_z1^z2 f(x,y,z) dx dy dz.

If x₁, x₂ are constants; y₁,y₂ are either constants or functions of x and z1, z2 are either constants or function of x and y, then this integral is evaluated as follows:

First f(x, y, z) is integrated w.r.t z between the limits z₁ and z₂ keeping x and y fixed. The resulting expression is integrated w.r.t y between the limits y₁ and y₂ keeping x constant. The results just obtained is finally integrated w.r.t x from x1 to x2.
            Thus I= ∫_x1^x2 ∫_y1(x)^y2(x) ∫_z1(x,y)^z2(x,y) f(x,y,z) dz dy dx

Here, the integration is carried out from the innermost rectangle to the outermost rectangle.


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