Let $B$ be the solid tetrahedron bounded by $x = 0, y = 0, z = 0$, and $x+y+z = 1$. Assume that the density of $B$ is given by the function $y$. Compute the mass of $B$.
I am having trouble solving this problem. Would I have to first find the integral of the tetrahedron given the bounds, but how would I do that without a function? Also, how would I determine the density of the tetrahedron? Thanks for your help!
$\endgroup$1 Answer
$\begingroup$$Volume = \int_0^1\int_0^{1-x}\int_0^{1-x-y} dz\ dy\ dx = \frac 16\\ Mass = \int_0^1\int_0^{1-x}\int_0^{1-x-y} \rho(x,y,z)\ dz\ dy\ dx\\$
and it is given that $\rho(x,y,z) = y$
$\int_0^1\int_0^{1-x}\int_0^{1-x-y} y\ dz\ dy\ dx\\ \int_0^1\int_0^{1-x} yz|_0^{1-x-y}\ dy\ dx\\ \int_0^1\int_0^{1-x} y(1-x) - y^2\ dy\ dx\\ \int_0^1 \frac 12 y^2(1-x) - \frac 13 y^3|_0^{1-x}\ dx\\ \int_0^1 \frac 16 (1-x)^3\ dx\\ -\frac 1{24} (1-x)^4|_0^1\\ \frac 1{24}$
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