Recently I was playing around with the IIS 7 PowerShell snap-in and I found a simple way of obtaining what application pool a given web application is running under.
The code is pretty simple and to do this, run from a standard PowerShell console:
##Import the IIS Admin module – you will need this!!!
##Get the Application Pool for a given website – Replace ‘Default Web Site’ with your site and you find what app pool the site is running under:
Get-Item "IIS:\Sites\Default Web Site"| Select-Object applicationPool
Although this isn't really SharePoint related, it will save you heaps of time rather than having to fire up the UI
More details can be found here at the following links: