Yeah you make sense. A connection with a dynamic IP isn't the best environment for hosting a site, not even close. You could set up a daemon to update the DNS server with your current IP every 5 mins but then with DNS propagation you'd probably have a new IP by the time the rest of the world could see your site. If you connect through a router and don't turn your router off then you keep your IP because the connection stays, I have a dynamic IP but it's been the same for weeks
Or you could try a service like DynDNS, for free they give you a program to run that updates their servers and you get a subdomain. I think for a small fee you can use your domain with them but I'm not sure, you could have a look.