## Triple Integral

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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(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 x

_{1}, x

_{2}are constants; y

_{1},y

_{2}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 z

_{1}and z

_{2}keeping x and y fixed. The resulting expression is integrated w.r.t y between the limits y

_{1}and y

_{2}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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