The discussion revolves around the calculation of work involving an exponential function, specifically using the equation W=∫0∞PdV where P is defined as P=e-v². Participants are exploring the integration of this function and the methods to evaluate the integral.
Some participants have provided guidance on the approach to take with the integral, suggesting the use of polar coordinates and referencing the Gaussian integral. There is an exploration of whether the interval from ∞ to 0 can be considered negligible, indicating a productive direction in the discussion.
Participants are navigating the complexities of integrating an exponential function and are considering the implications of the limits of integration in their calculations. The discussion reflects a learning process around the Gaussian integral and its application to the problem at hand.
kabailey said:W=∫0∞PdV
P=e-v2
Do I simply substitue P in the original equation, differentiate, then integrate?
W=∫0∞e-v2dV
W=∫0∞(e-v2)/2v
So far is this correct?
LCKurtz said:No, it isn't. That integral is usually worked by a trick of multiplying it by itself and changing it to polar coordinates. See
http://en.wikipedia.org/wiki/Gaussian_integral
There they do it from ##-\infty## to ##\infty##, but the same idea applies.
kabailey said:I was not familiar with the Gaussian intergral. Well since it's only half the distance I would divide by 2 ∴
W=∫0∞e-v2dV
=(√∏)/2|∞0
so does that mean that the interval of ∞ to 0 is negligible?
kabailey said:thank you for your help. my final answer is W=(√∏)/2