37.7k views
4 votes
Let r be the region enclosed by y = x⁴, x = 2, and y = 0 rotated about the y-axis. What is the resulting solid?

1) A cone
2) A cylinder
3) A sphere
4) A torus

1 Answer

4 votes

Final answer:

The solid of revolution formed by rotating the region bounded by y = x⁴, x = 2, and y = 0 around the y-axis does not perfectly match any standard geometrical figure but is closest to a torus, which is not entirely accurate. The volume of a sphere is ⅓πr³, which is distinct from the surface area, 4πr².

Step-by-step explanation:

The student is inquiring about the resulting solid of a region enclosed by certain equations, which is then rotated around the y-axis. To clarify, when a region bounded by the curve y = x⁴, the line x = 2, and the x-axis (y = 0) is rotated about the y-axis, it creates a three-dimensional solid known as a solid of revolution. In this case, the resulting figure is not a cone, a cylinder, or a sphere; instead, the shape starts as a point at the origin and widens as it extends outwards, forming a curved, bell-shaped object that is neither of the given standard geometrical figures. Therefore, based on the options provided, the closest description, although not entirely accurate, would be a torus, even though a genuine torus would require the revolution of a circular area rather than a power function like x⁴.

The volume of a sphere, for reference, is given by the formula ⅓π, not 4². The former expression relates to the volume, while 4π represents the surface area of a sphere. This distinction is crucial when studying geometric properties such as volume and surface area.

User Dan Gerhardsson
by
8.2k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.