Find the mass of the solid that occupies the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane z = 1 and has density function $\rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .
A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F)$\pi$

G) $\frac { 4 \pi } { 15 }$

H)2$\pi$

Question

Find the mass of the solid that occupies the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane z = 1 and has density function $\rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .
A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$
  
F)$\pi$

G) $\frac { 4 \pi } { 15 }$

H)2$\pi$

Quizplus · Accepted Answer

The mass of the solid can be found by integrating the density function over the volume of the solid. The solid is bounded by the paraboloid $z = x^2 + y^2$ and the plane $z = 1$, which means it is a solid of revolution and its cross-section is a circle with radius 1 (since $z = 1$ when $x^2 + y^2 = 1$). The density function $\rho(x, y, z) = \sqrt{x^2 + y^2}$ suggests cylindrical symmetry, so we can use cylindrical coordinates for the integration. The mass $M$ is given by the triple integral $\int\int\int_V \rho(x, y, z) dV$, where $V$ is the volume of the solid. In cylindrical coordinates, this becomes $\int_0^{2\pi} \int_0^1 \int_{r^2}^1 r^2 dz dr d\theta$, which simplifies to $\frac{4\pi}{15}$.

Find the Mass of the Solid That Occupies the Region z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2

Find the Mass of the Solid That Occupies the Region $z = x ^ { 2 } + y ^ { 2 }$