Volume of Revolution about the y-axis generated between curves

Volume of Revolution about the y-axis between Curves

The Volume of Revolution is the volume of a solid formed when a planar region is rotated about a line lying on the same plane.
The volume of the solid of revolution generated by rotating an area bound between two curves about the y-axis is calculated using definite integrals.

Suppose that y=f(x) and y=g(x) are two curves, and let R be the region that lies between these two curves from x=a to x=b. If R is revolved about the y-axis, then it sweeps out a solid.
To calculate the volume of this solid, we consider a typical small element, a thin washer with outer radius f(x), inner radius g(x), and thickness dx.
The volume element of this washer is essentially a thin cylindrical shell with volume dV = 2πx(y_2 - y_1) dx, where y_2 = f(x) and y_1 = g(x). This formula represents the difference of two circular cylindrical shells and determines the volume of the thin washer.
To find the volume of the solid, we then integrate these volume elements over the interval [a, b], i.e., V = ∫ from a to b of (2πx(f(x) - g(x)) dx.

The integrand function must be continuous on the interval [a, b]. This ensures that all slices or washers are well-defined.
The curves y=f(x) and y=g(x) need not cross, but the lower limit of the integration, a, should be less than or equal to the point where the curves intersect in the xy-plane, and the upper limit of the integration, b, should be greater than or equal to this point.
It’s important to note that f(x) > g(x) for all x in [a, b] if the region R is to be ‘above’ the x–-axis.

The Volume of Revolution concept is used in several areas of science and engineering. For instance, it serves as a mathematical model in fluid dynamics to describe the flow of fluids around objects.
An everyday example is the design of objects with rotational symmetry, such as containers or vases, where the artist or engineer might calculate the volume of material needed or the capacity of the object.