How to evaluate:
∫∫∫ xyz/(x^2+y^2)^1/2 dv
E
where E is the solid "quarter-cylinder" in the first octant bounded by a cylinder of radius a>o with axis long the z axis, the planes y=0,x=0,z=0 and z=b, wiht b>0.
Thankyou very much!
How to evaluate:
∫∫∫ xyz/(x^2+y^2)^1/2 dv
E
where E is the solid "quarter-cylinder" in the first octant bounded by a cylinder of radius a>o with axis long the z axis, the planes y=0,x=0,z=0 and z=b, wiht b>0.
Thankyou very much!
The best way to tackle this one is to change to cylindrical coordinates. So we'd get:
$\displaystyle
\int_0^b \int_0^{\pi /2} \int_0^a \frac{(\rho cos\phi )(\rho sin\phi ) z}{\rho } \rho d\rho d\phi dz
$
$\displaystyle
\int_0^b zdz \int_0^{\pi /2} cos\phi sin\phi d\phi \int_0^a \rho ^2 d\rho
$
$\displaystyle
(\frac{b^2}{2})(\frac{1}{2})(\frac{a^3}{3})
$
$\displaystyle
\frac{a^3 b^2}{12}
$