Volumes of revolution around the y-axis

To find the volume of a solid of revolution generated by rotating a curve around the y-axis between two points, integrate from the lower to the upper limit the square of the given function with respect to y, multiplied by pi.
The formula for finding a volume of revolution generated by rotating a curve y=f(x) around the y-axis from y=c to y=d is given by V = π∫[x²]dy evaluated from c to d.
This formula comes from considering infinitesimal cylindrical slices of the solid, each of which has a volume of πx²d(y), where x is the radius of the cylinder and dy is its height.
Here x² becomes the equivalent of a radius squared, the diameter of the infinitesimally small discs that make up the solid.
Understand that the infinitesimal volume is circular, hence the π in the equation. The variable y helps determine the limits of integration.
Always draw a diagram. Visualising the volume created by the revolution of the curve around the axis helps enormously.
The position of the curve in relation to the axis of rotation can change the method needed to compute the volume of the solid generated.
Be familiar with the difference between rotating about the y-axis and the x-axis. The formula used and the method of calculating volumes is notably different.
Practical application: Solids of revolution frequently appear in real-world objects, hence understanding their volume and knowing how to calculate them is highly valuable for future applications.
Practice with different functions. Functions of x that are powers, exponentials, or trigonometric functions provide a wide array of practice examples.
Frequently re-evaluating understanding of this topic by doing variety of practice problems and turning theoretical knowledge into practical skills is highly beneficial.
If the integrals prove to be too difficult to evaluate analytically, techniques such as numerical integration may be used.
This topic links with other areas in Further Mathematics. For example, it provides practical applications of integration, which is a key topic in calculus. Connecting this knowledge with other topics will reinforce understanding.
Be aware of the potential for examination questions to combine this topic with others, such as finding the centroid or the surface area of a solid of revolution.