my.domain.name.de links to the webservermy.domain.name.de/shiny links to the shiny server<VirtualHost *:80>
ServerAdmin ...
ServerName my.domain.name.de
<Proxy /shiny/>
Allow from localhost
</Proxy>
ProxyPreserveHost On
ProxyPass /shiny/ http://localhost:3838/
ProxyPassReverse /shiny/ http://localhost:3838/
DocumentRoot /var/www
<Directory />
...
</Directory>
...
</VirtualHost>
Have you ever been able to create an apache virtual host? (Ie two hosts with different names pointing to the same IP but serving different content). I guess, this is probably where you want to start. There is a very good how to in the apache doc. But first before starting to tweak your server, you need to have two DNS entries pointing to the same adress (usually one A record and then a bunch of CNAME or ALIAS records).
Not sure at what level you are but if nothing of this make sense, you have a bit of reading before trying to implement a proxy server.
Let us know, we can probably help you or show you where to find Information.
RedirectMatch permanent ^/test$ /test/