Washer Method

The volume of the solid formed by rotating the area between the curves of and and the lines and about the x-axis is given by

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

Shell integrationThe cylinder method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.

The volume of the solid formed by rotating the area between the curves of and and the lines and about the y-axis is given by

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

Parametric form

When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the x-axis or the y-axis are given by

Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the x-axis or the y-axis are given by