Surface tension gases. The pressure inside an ideal (one surface) soap bubble can be derived from thermodynamic free energy considerations. At constant temperature and particle number, , the differential Helmholtz energy is given by where is the difference in pressure inside and outside of the bubble, and is the surface tension. In equilibrium, , and so, For a spherical bubble, the volume and surface area are given simply by and Substituting these relations into the previous expression, we find which is equivalent to the Young–Laplace equation when . For real soap bubbles, the pressure is doubled due to the presence of two
Solid of revolution suitable integral. The disk method is used when the slice that was drawn is "perpendicular to" the axis of revolution; i.e. when integrating "parallel 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 -axis is given by If (e.g. revolving an area between the curve and the -axis), this reduces to: The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in