Volume of revolution

Answer 1

The volume of revolution is the volume generated when a two-dimensional shape is rotated around a given axis. This process is also known as finding the volume of a solid of revolution. The two-dimensional shape can be a curve, a region bounded by curves, or a solid region. The axis of revolution can be any line in the plane, but it is most commonly the x-axis or y-axis.

To find the volume of revolution, we need to use integration. The basic idea is to slice the solid into thin disks perpendicular to the axis of revolution, and then sum the volumes of these disks using integration.

For example, let's consider rotating the curve y = x^2 around the x-axis from x = 0 to x = 1. To find the volume of revolution, we can slice the solid into thin disks of thickness dx, with radius y = x^2. The volume of each disk is πy^2 dx, so the total volume is:

V = ∫0^1 πy^2 dx

V = ∫0^1 πx^4 dx

V = π/5

So the volume of the solid of revolution is π/5 cubic units.

Note that the volume of revolution can also be calculated using the disk method or the washer method. These methods involve slicing the solid into disks or washers perpendicular to the axis of revolution, and then summing their volumes using integration.