If lets say I have $\displaystyle y=x^{2}$ and $\displaystyle x=2$ as the area (top view) of a shape and the height is 10 units.
How would I find the x,y and z moments of this 3d object? Also how would I find the center of mass?
If lets say I have $\displaystyle y=x^{2}$ and $\displaystyle x=2$ as the area (top view) of a shape and the height is 10 units.
How would I find the x,y and z moments of this 3d object? Also how would I find the center of mass?
The (first) moment of inertia in space is.
$\displaystyle M_{xy} = \iiint_V \delta(x,y,z)z dV$
$\displaystyle M_{yz} = \iiint_V \delta(x,y,z)xdV$
$\displaystyle M_{xz} = \iiint_V \delta(x,y,z) ydV$
Where $\displaystyle \delta(x,y,z)$ is the density of the region.
no, $\displaystyle \delta (x,y,z)$ is the density function. it depends on what the question says it is. the equations you were given describe the solid over which you are integrating, not the density of the solid, we need additional conditions to find $\displaystyle \delta (x,y,z)$
also, define $\displaystyle m = \iiint_V \delta(x,y,z) dV$
the center of mass is the point $\displaystyle (\bar {x}, \bar {y}, \bar {z})$
where $\displaystyle \bar {x} = \frac {M_{yz}}{m}$, $\displaystyle \bar {y} = \frac {M_{xz}}{m}$, and $\displaystyle \bar {z} = \frac {M_{xy}}{m}$