Help With Integration Calculation - Struggling?

[JWelford]

\int_{-0.4088}^{-\infty}\,e^{-1/2.4^2}d struggling to solve this calculation. Not sure if i have written the formula in the right way. First post on this site. thanks

 

[topsquark]

\int_{-0.4088}^{-\infty}\,e^{-1/2.4^2}d struggling to solve this calculation. Not sure if i have written the formula in the right way. First post on this site. thanks

\(\displaystyle \int_{-.4088}^{\infty} e^{-1/2.4} d\)

There needs to be a variable in there somewhere!

-Dan

 

[JWelford]

\(\displaystyle \int_{-.4088}^{\infty} e^{-1/2.4} d\)

There needs to be a variable in there somewhere!

-Dan

woops its e^-1/2 . u^2 du

and the lower bound is minus infinity

 

[topsquark]

woops its e^-1/2 . u^2 du

and the lower bound is minus infinity

Here's a trick to remember. Let
\(\displaystyle I = \int_{-\infty}^{\infty} e^{-u^2/2}~du\)

Now, u is a "dummy variable" so we can just as easily say that \(\displaystyle I = \int_{-\infty}^{\infty} e^{-v^2/2}~dv\).

Since u and v are unrelated we can multiply these together:
\(\displaystyle I^2 = \left ( \int_{-\infty}^{\infty} e^{-u^2/2}~du \right ) \left ( \int_{-\infty}^{\infty} e^{-v^2/2}~dv \right ) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-u^2/2} ~ e^{-v^2/2}~dv~du\)

So
\(\displaystyle I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{(-1/2)(u^2 + v^2)} ~dv~du\)

Define a set of polar coordinates \(\displaystyle ( r, \theta )\) such that \(\displaystyle u = r~cos( \theta )\) and \(\displaystyle v = r~sin( \theta )\). The Jacobian is equal to r and \(\displaystyle u^2 + v^2 = r^2\), so
\(\displaystyle I^2 = \int_{0}^{\infty} \int_{0}^{2 \pi} e^{-r^2/2} ~r~d \theta~dr\)

The rest I leave to you.

-Dan