Volume of revolution
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.
The volume of revolution refers to the volume of a solid obtained by rotating a 2D shape about an axis of rotation. The method used to determine the volume depends on the shape and the axis of rotation. There are two common methods: the disc method and the shell method.
1. Disc Method:
- Consider a 2D shape, such as a region bounded by a curve y = f(x), the x-axis, and two vertical lines x = a and x = b.
- To find the volume of revolution, you need to rotate this region about the x-axis.
- Divide the interval [a, b] into small subintervals and consider a typical subinterval [x, x + ∆x].
- Construct a disc or washer (if the region has a hole) with radius f(x) and thickness ∆x.
- The volume of each disc or washer is π(f(x))^2∆x.
- Sum up the volumes of all the discs or washers using integration: V = ∫[a,b] π(f(x))^2 dx.
2. Shell Method:
- Consider a 2D shape bounded by a curve y = f(x), the x-axis, and two vertical lines x = a and x = b.
- To find the volume of revolution, you need to rotate this region about a vertical line x = c.
- Divide the interval [a, b] into small subintervals and consider a typical subinterval [x, x + ∆x].
- Construct a cylindrical shell with radius x - c, height f(x), and thickness ∆x.
- The volume of each shell is the lateral surface area multiplied by ∆x: V = 2π(f(x))(x - c)∆x.
- Sum up the volumes of all the shells using integration: V = ∫[a,b] 2π(f(x))(x - c) dx.
Note: The choice between the disc method and shell method depends on the shape of the region and the axis of rotation. It is important to identify the appropriate method based on the given problem.