Volume of revolution using integration
Volume of revolution is a method in integral calculus used to find the volume of a solid that is created by rotating a two-dimensional region about an axis. This process is also known as finding the volume of a solid of revolution. The idea is to use the concept of definite integral to find the volume of the solid by adding up the volumes of infinitesimal disks (or washers) that make up the solid.
To find the volume of revolution, we need to:
- Identify the region that we want to rotate, and the axis of rotation.
- Choose a coordinate system and integrate the cross-sectional area of the region along the axis of rotation.
- Use the definite integral to find the total volume of the solid by adding up the volumes of the infinitesimal disks.
There are two main methods to find the volume of revolution: disk method and washer method. In the disk method, we find the volume of the solid by summing up the volumes of circular disks, while in the washer method, we find the volume of the solid by summing up the volumes of circular washers (which are circular disks with holes in the center).
Example: Suppose we want to find the volume of the solid that is created by rotating the region bounded by the graph of y = x^2 and the x-axis about the x-axis.
Using the disk method, we can find the volume of the solid by summing up the volumes of infinitesimal disks. To do this, we choose a coordinate system, and then integrate the cross-sectional area of the region along the axis of rotation.
The cross-sectional area of each disk is given by:
A = πr^2
Where r is the radius of the disk. To find r, we need to find the value of y for a given value of x. From the equation y = x^2, we have:
r = x^2
So, the cross-sectional area of each disk is:
A = πr^2 = πx^4
To find the volume of the solid, we need to integrate the cross-sectional area of the region along the axis of rotation, using the definite integral:
V = ∫_{-a}^a πx^4 dx
Where a is the limit of integration, which is the maximum value of x. To find the exact value of the definite integral, we need to use proper methods of integration.
In this example, we used the disk method to find the volume of a solid of revolution, by integrating the cross-sectional area of the region along the axis of rotation.