Find the mass of the solid $S$ bounded by the paraboloid $z = 6 x ^ { 2 } + 6 y ^ { 2 }$ and the plane $z = 5$ if $S$ has constant density 3 . Select the correct answer.
A) $16.25$
B) $15.07$
C) $24.91$
D) $13.92$
E) $19.63$

Question

Quizplus · Accepted Answer

The mass of the solid is found by integrating the density over the volume of the solid. The solid is bounded by the paraboloid $z = 6x^2 + 6y^2$ and the plane $z = 5$. Converting to polar coordinates, the region of integration is a circle of radius $r = \sqrt{\frac{5}{6}}$. The mass is given by the integral $3 \int_0^{2\pi} \int_0^{\sqrt{\frac{5}{6}}} (5 - 6r^2) r \, dr \, d\theta$, which evaluates to 19.63.

Find the Mass of the Solid SSS Bounded by the Paraboloid z=6x2+6y2z = 6 x ^ { 2 } + 6 y ^ { 2 }z=6x2+6y2

Find the Mass of the Solid $S$ Bounded by the Paraboloid $z = 6 x ^ { 2 } + 6 y ^ { 2 }$