Stand next to the switch, then watch the spinning disc at the end of the room. When you get to the end, time your jump through the flames carefully. That makes it so you can follow the spiraling path forward safely. Use your map to switch the direction that the driveshaft is spinning - this will make the spike-covered cylinder spin the other way. You want the cylinder to be spinning from left to right. Head inside the room and stop in front of the spinning, spike-covered cylinder.
Hop off, then wait at the giant spinning disc until a door spins past. Run over and let it carry you up to the platform on the right. The gear ahead of you on the right has paddle-like platforms coming up out of the lava that you can ride. From the map terminal, head back to the entrance.
