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
Gabriel's Horn using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between and , where . Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume and the surface area : The surface area formula above gives a lower bound for the area as 2 times the natural logarithm of . There is no upper bound for the natural logarithm of as approaches infinity. That means, in this case, that the horn has an infinite surface area. That