Find the volume of the solid bounded by the paraboloid z = 16 - x² - 4y² and the planez = 0. A) 48$\pi$ cubic units B) 64$\pi$ cubic units C) 96$\pi$ cubic units D) 128$\pi$ cubic units E) 81$\pi$ cubic units

Question

Quizplus · Accepted Answer

The volume of the solid can be found by integrating the given paraboloid equation $z = 16 - x^2 - 4y^2$ over the region it covers above the $z=0$ plane. The equation of the paraboloid can be rearranged to find the boundary of the solid in the xy-plane: $x^2 + 4y^2 = 16$. This represents an ellipse, but since it's in the form $x^2 + 4y^2 = 16$, it can be simplified to a circle by substituting $y = \frac{y'}{2}$, making the equation $x^2 + y'^2 = 16$, which is a circle of radius 4 centered at the origin. The volume can be calculated using cylindrical coordinates, where $r^2 = x^2 + y^2$, and the limits for $r$ are from 0 to 4, and for $\theta$ from 0 to $2\pi$. The integral becomes $\int_0^{2\pi} \int_0^4 (16 - r^2) r dr d\theta$. Solving this integral gives the volume of the solid as $64\pi$ cubic units.

Find the Volume of the Solid Bounded by the Paraboloid π\piπ

Find the Volume of the Solid Bounded by the Paraboloid $\pi$