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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(xr, yr,zr,) δVr
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
        ∫ ∫ ∫x1x2 ∫ ∫ ∫y1y∫ ∫ ∫z1z2 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= ∫x1x2 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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