Find the volume of the given solid by transforming to a double integral in polar coordinates.The solid is bounded by x² + y² = 36, z = 0, and z = x² + y². A) 618$\pi$cubic units B) 628$\pi$ cubic units C) 638$\pi$ cubic units D) 648$\pi$ cubic units E) 658$\pi$ cubic units

Question

Quizplus · Accepted Answer

The given solid is bounded by the cylinder $x^2 + y^2 = 36$, the plane $z = 0$, and the cone $z = \sqrt{x^2 + y^2}$. To find the volume using polar coordinates, we convert the equations: $x^2 + y^2 = 36$ becomes $r^2 = 36$ (or $r = 6$), and $z = \sqrt{x^2 + y^2}$ becomes $z = r$. The volume can be found by integrating in polar coordinates with the limits for $r$ from 0 to 6 and $\theta$ from 0 to $2\pi$, and $z$ from 0 to $r$. The double integral becomes $\int_{0}^{2\pi} \int_{0}^{6} r \, dr \, d\theta$. Solving this gives the volume as $648\pi$ cubic units.

Find the Volume of the Given Solid by Transforming to a Double