Find the volume of the solid lying inside the cylinder x² + y² = 4, above the plane z = x - y - 8, and below the surface z = 8 - x³. A) 72$\pi$ cubic units B) 64$\pi$ cubic units C) 48$\pi$ cubic units D) 32$\pi$ cubic units E) 56$\pi$ cubic units

Question

Quizplus · Accepted Answer

The given cylinder equation is $x^2 + y^2 = 4$, which is a cylinder of radius 2 centered on the z-axis. The top surface is $z = 8 - x^2$, and the bottom surface is $z = x - y - 8$. To find the volume of the solid, we integrate over the area of the circle (base of the cylinder) and between the two z-values (top and bottom surfaces). The volume can be found using cylindrical coordinates, where the radius $r$ goes from 0 to 2, and the angle $\theta$ goes from 0 to $2\pi$. The integral setup would involve the difference between the top and bottom surfaces, integrated over the area of the circle. The correct answer, 64$\pi$ cubic units, suggests that the calculation accounts for the entire volume enclosed by these boundaries, considering the symmetry and shape of the cylinder and the given surfaces.

Find the Volume of the Solid Lying Inside the Cylinder π\piπ

Find the Volume of the Solid Lying Inside the Cylinder $\pi$