Modelling with volumes of revolution

A volume of revolution is formed by taking a two-dimensional region and revolving it around an axis. The axis can be any straight line, not necessarily part of the coordinate system.
Assuming the two-dimensional region creates a distance, r(x), perpendicular to the axis of revolution, the volume, V, of the solid of revolution is determined by the integral ∫π[r(x)]^2 dx, where the integral is performed over the range of x.
When the x-axis is the axis of revolution, r(x) is the positive y-value for the curve above the x-axis. However, when the y-axis is the axis of revolution, r(x) should have the horizontal distance from the y-axis.
Be cautious when the function crosses over the axis around which you are rotating. In such cases, it may be easier to calculate separate volumes of rotation for different intervals of x and then sum those volumes to get the total.
Understand the Disc Method, which involves slicing up the shape into infinitesimally thin discs parallel to the axis of revolution. The volume of each disc is pi times the square of the radius times the thickness. Summing the volumes of these disks across the whole shape gives the total volume.
Understand the Shell Method, which is applicable when it is easier to slice the shape into cylindrical shells. The volume of each shell is 2π times the radius times the height times the thickness. Summing these volumes gives the total volume.
When forming solids of revolution about the y-axis, remember to express the function and bounds of integration in terms of y rather than x.
Practice problems such as rotating the area under a function about the x-axis, rotating the area between two functions about the y-axis, or rotating areas bounded by a piecewise function.
Understand the real-world applications of volumes of revolution, including determining volumes of shaped objects, analysis of stress-strain in mechanical and civil engineering, fluid flow in physics, and calculating volumes in medical imaging technologies.
Make sure to check your answers using geometry where possible. For example, rotating a rectangle about one side would make a cylindrical shape where volume could be confirmed using the base times height formula.
Problems can often be solved either by the Disc Method or Shell Method. However, recognizing which method simplifies calculations is an essential skill to develop.
Lastly, always remember to give your final answer with correct units of volume.

Remember to practice various examples of modelling with volumes of revolution to become adept in using both techniques and solving complex problems. Visualizing the shapes formed can also be very helpful.