If $\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ is the volume density of the solid between two concentric spheres with radii 1 and 2, then its mass is

A) $18 \pi$
B) $15 \pi$
C) $12 \pi$
D) $8 \pi$
E) $6 \pi$

Question

If $\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ is the volume density of the solid between two concentric spheres with radii 1 and 2, then its mass is

A) $18 \pi$
B) $15 \pi$
C) $12 \pi$
D) $8 \pi$
E) $6 \pi$

Quizplus · Accepted Answer

The mass is calculated by integrating the density function over the volume of the region between the two spheres. The density function $\sigma(x, y, z) = \sqrt{x^2 + y^2 + z^2}$ is the radial distance $r$ in spherical coordinates. The mass is given by the integral $\int_{1}^{2} \int_{0}^{\pi} \int_{0}^{2\pi} r \cdot r^2 \sin \theta \, d\phi \, d\theta \, dr$, which evaluates to $15\pi$.

If σ(x,y,z)=x2+y2+z2\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }σ(x,y,z)=x2+y2+z2​

If $\sigma ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$