Volumes of revolution around the x-axis

A volume of revolution is created when a 2D shape is rotated a full 360 degrees about an axis. In the case of rotation about the x-axis, y = f(x) is the function of the curve being rotated.
The method for finding the volume of revolution uses the principle of integration. The integral symbol can be thought of as summing up all the infinitesimally small disks that make up the solid. The thickness of each disk is dx, a tiny change in x.
The formula of the volume of revolution about the x-axis is V = π ∫[a, b] (f(x))^2 dx. Here, ‘a’ and ‘b’ represent the points that bound the interval on the x-axis over which the rotation is happening.
In the formula, (f(x))^2 gives the square of the function which, when multiplied by π, helps us get the area of the circle at that particular x value. We then integrate over the interval [a, b] to sum up all these infinitesimally small area elements to get the total volume.
For an example, consider the straight line y = x from x = 0 to x = 1. To find the volume of the solid created when this region is rotated about the x-axis, use the volume of revolution formula V = π ∫[0, 1] (f(x))^2 dx, where f(x) = x.
Don’t forget to square the function before integrating when applying this formula. It corresponds to the calculation of the area of the circular disc at any position x, which contributes to the total volume.
Differentiation or anti-differentiation may be necessary for the function f(x) before integration can occur. Also, sometimes it may be necessary to split the integral into more manageable sections.
The concept of volumes of revolution can be applied to find the volume of various shapes in real-world situations such as the volume of a dome, flower vase, or any other object which has rotational symmetry around the x-axis.
Practice with a variety of functions for f(x), including polynomials, and simple trigonometric and exponential functions, to become comfortable with different methods of integration and their application in volume of revolution problems.
Carefully sketch the 2D area that is being revolved around the x-axis, include the bounding area, this can often help you visualize the process and understand the problem.
When the problem involves a bounded area between two curves, be certain you are integrating the difference of the two functions, essentially you are finding the ‘net’ area being revolved to find the volume.
As with all mathematical methods, practice consistently and review your work critically to understand and rectify any errors. This will help further develop your understanding and application of the concept of volumes of revolution around the x-axis.