Question

Find the mass of the triangle with vertices (1; 0; 0), (0; 2; 0), (0; 1; 1) if the density function is given by (x; y; z)

Answer 1

`=(17)/(3)`

Explanation:

Density of a function is
`\rho(x,y)=x^(2) +y^(2)`

I have drawn the right angle triangle for visualization

equation for the line passing through (1,0) and (0,4) is

`(y)/(4) =(x-1)/(-1)`

`y=4-4x`

=
`$$\int_(0)^1\int_(0)^(4-4x) (x^(2)+y^(2))dydx$$`

=
`$\int_(0)^(1)(x^(2)+(y^(3) )/(3))dx$`

=
`$\int_(0)^(1)(x^(2)(4-4x)+(1)/(3)(4-4x)^(3) )dx$`

=
`$\int_(0)^(1)((-76)/(3)x^(3) +68x^(2)-64x^{}+(64)/(3)  )dx$`

=
`=((-76)/(3) (x^(4) )/(4))+(68)/(3)x^(3)-32x^(2)+(64)/(3)x^{})`

`=(17)/(3)`