The Method of Mechanical Theorems convenient to use a little bit of coordinate geometry. If a sphere of radius 1 is placed with its center at "x" = 1, the vertical cross sectional radius formula_2 at any "x" between 0 and 2 is given by the following formula: The mass of this cross section, for purposes of balancing on a lever, is proportional to the area: Archimedes then considered rotating the triangular region between "y" = 0 and "y" = "x" and "x" = "2" on the "x"-"y" plane around the "x"-axis, to form a cone. The cross section of this cone is a circle
Feynman–Kac formula term contains, in parentheses, the above partial differential equation and is therefore zero. What remains is Integrating this equation from "t" to "T", one concludes that Upon taking expectations, conditioned on "X" = "x", and observing that the right side is an Itô integral, which has expectation zero, it follows that The desired result is obtained by observing that and finally In quantitative finance, the Feynman–Kac formula is used to efficiently calculate solutions to the Black–Scholes equation to price options on stocks. Feynman–Kac formula The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial