# Math Help - Triple Integral

1. ## Triple Integral

Evaluate $\int\int_E\int \sqrt{x^2+y^2+z^2}\;dV$where E is the solid bounded below by the conic surface $3z^2=x^2+y^2,z\ge 0$ and bounded above by the spherical surface $x^2+y^2+z^2=5$.

Thanks

2. Originally Posted by polymerase
Evaluate $\int\int_E\int \sqrt{x^2+y^2+z^2}\;dV$where E is the solid bounded below by the conic surface $3z^2=x^2+y^2,z\ge 0$ and bounded above by the spherical surface $x^2+y^2+z^2=5$.

Thanks
The upper surface is $z = \sqrt{5-x^2-y^2}$ and lower surface is $z = \frac{1}{\sqrt{3}}\sqrt{x^2+y^2}$. You need to know the region over which you are integrating. Note if $x^2+y^2+z^2 = 5 \implies 3x^2+3y^2+3z^2 = 15$ so $4x^2+4y^2 = 15\implies x^2+y^2 = \frac{15}{4}$.

Thus, we get, $\iint_A \int_{(1/3)\sqrt{x^2+y^2}}^{\sqrt{5-x^2-y^2}} f(x,y,z) dz dA$ where $A$ is the disk $x^2+y^2 \leq \frac{15}{4}$.

Now use Polar change of variable.