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.
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 paraboloid $z = 6x^2 + 6y^2$ intersects the plane $z = 5$ at $6x^2 + 6y^2 = 5$, or $x^2 + y^2 = \frac{5}{6}$. The volume integral in cylindrical coordinates is $\int_0^{2\pi} \int_0^{\sqrt{5/6}} \int_{6r^2}^{5} 3 \, dz \, r \, dr \, d\theta$. Solving this gives the mass as 19.63.

Find the Mass of the Solid S Bounded by the Paraboloid