Update: I have posted another answer that works with the Spreadsheets v4 API.
Old Answer:
The answer from Alain is correct, but you also need to set the gid=parameter to specify which worksheet to export.
For example, if your ‘application/pdf’ export link is like this:
docs.google.com/feeds/download/spreadsheets/Export?key=<FILE_ID>&exportFormat=pdf
You can just change it to this to download the first worksheet:
docs.google.com/feeds/download/spreadsheets/Export?key<FILE_ID>&exportFormat=csv&gid=0
There is a bit of a problem, though as there is no reliable way to get the gid for a given worksheet through the API and they are not zero based indexes. If you delete a worksheet, that gid does not get reused. You can see the gid in the URL in your browser though, so if your worksheet information is constant you can just get that from there. See http://code.google.com/a/google.com/p/apps-api-issues/issues/detail?id=1813 and http://code.google.com/a/google.com/p/apps-api-issues/issues/detail?id=3240 for more info on that problem.