Evaluate `int int int (x+y+z)` `dxdydz ` over the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1.

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#### Solution

`I=int_(x=0)^1 int_(y=0)^(1-x) int_(z=0)^(1-x-y) (x+y+z)dzdydx`

`I= int_(x=0)^1 int_(y=0)^(1-x) [(x+y+z)^2/2]^(1-x-y) dydz`

`I=1/2 int_(x=0)^1 int_(y=0)^(1-x) [1-(x-y)^2]dydx`

`I=1/2 int_(x=0)^1 [y-(x+y)^2/2]^(1-x) dx`

`I=1/2int_(x=0)^1 [(1-x)-1/3+x^3/3]dx`

`I=1/2 [(2x)/3+x^2/2+x^4/12]_0^1`

`I= 1/2. 3/12=1/8`

∴`I=1/8`

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function

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