Volumes of revolution around the y-axis

First understand the concept: when a curve is rotated around the y-axis, it sweeps out a volume in 3D space. This is known as the volume of revolution.
The general formula for calculating the volume of revolution around the y-axis is ∫ (from a to b) of [x^2 * f(x) dx]. The integration is taken between two points a and b on the y-axis, f(x) is the y-value of the function being revolved, and x represents the distance from the y-axis.
To apply the formula, you must be able to express x as a function of y. If the equation of the curve is given as y = f(x), you’ll need to rearrange it to x = g(y) in order to plug into the formula.
If the region being revolved doesn’t start at the y-axis, you need to adjust the formula. The adjustment subtracts a smaller volume of revolution whose radius is the distance between the y-axis and the starting point of the region.
To solve these integral problems, note the cross-sectional area formula, which is a function of y and then integrate this from a to b on the y-axis.
In some cases, your integration limits may be given in terms of x instead of y. In these cases, you will need to convert these limits to y values before performing the integration.
The complexity of integration could vary depending on the function given. It could involve simple, trigonometric or even transcendent functions. Review and master different techniques of integration to solve these problems effectively.
Remember that like all integral calculations, if the function being integrated is not continuous over the interval from a to b, the integral will need to be split into several distinct integrals over intervals where the function is continuous.
Practice is key. Work through plenty of examples where the function, limits of integration, or both are complex and require a variety of strategies to solve them. From these, learn to identify patterns and shortcuts that can simplify future calculations.