The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the x-axis is given by

If g(x) = 0 (e.g. revolving an area between the curve and the x-axis), this reduces to:

The method can be visualized by considering a thin horizontal rectangle at y between f(y) on top and g(y) on the bottom, and revolving it about the y-axis; it forms a ring (or disc in the case that g(y) = 0), with outer radius f(y) and inner radius g(y). The area of a ring is π(R² − r²), where R is the outer radius (in this case f(y)), and r is the inner radius (in this case g(y)). The volume of each infinitesimal disc is therefore πf(y)² dy. The limit of the Riemann sum of the volumes of the discs between a and b becomes integral (1).

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the y-axis is given by

If g(x) = 0 (e.g. revolving an area between curve and y-axis), this reduces to:

The method can be visualized by considering a thin vertical rectangle at x with height f(x) − g(x), and revolving it about the y-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is 2πrh, where r is the radius (in this case x), and h is the height (in this case f(x) − g(x)). Summing up all of the surface areas along the interval gives the total volume.

, 

and the y-axis about the y-axis. 

 

The area of the disk is then,

 

 

For the limits on the integral we can see that the “first” disk in the solid would occur at

and the “last” disk would occur at

.  Our limits are then :

.

 

The volume is then,

 

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