Evaluate the triple integral where E is the solid tetrahedro

Evaluate the triple integral, where E is the solid tetrahedron with vertices...

USE IMAGE BELOW TO SOLVE.

Evaluate the triple integral integralintegralintegral_E xydV where E is the solid tetrahedon with vertices (0,0,0),(4,0,0),(0,9,0),(0,0,5).

Solution

equation of plane is x/4 +y/9 +z/5 =1

change of variables :

x =4u ,y =9v, z =5w

u+v+w =1

jacobian =4*9*5 =180

_E xy dv

=_{[0 to 1][0 to 1-u][0 to 1-v-u]} 4u *9v *180 dw dv du

=6480_{[0 to 1][0 to 1-u][0 to 1-v-u]} u v dw dv du

=6480_{[0 to 1][0 to 1-u][0 to 1-v-u]} u v w dv du

=6480_{[0 to 1][0 to 1-u]} u v (1-v-u) dv du

=6480_{[0 to 1][0 to 1-u]} (uv-uv²-u²v) dv du

=6480_{[0 to 1][0 to 1-u]} ((1/2)uv² -(1/3)uv³-(1/2)u²v²) du

=6480_{[0 to 1]} ((1/2)u(1-u)² -(1/3)u(1-u)³-(1/2)u²(1-u)²) du

=6480_{[0 to 1]}-(1/6)u⁴+(1/2)u³-(1/2)u²+ (1/6)u du

=6480_{[0 to 1]}-(1/30)u⁵+(1/8)u⁴-(1/6)u³+ (1/12)u²

=6480[-(1/30)1⁵+(1/8)1⁴-(1/6)1³+ (1/12)1² ]

=6480[-(1/30)+(1/8)-(1/6)+ (1/12) ]

=54

_E xy dv=54