The air filter box is located under the broad flat vent tube at the right side of the engine bay.

The stock air box on a Mazda 3 has several components over it. We'll have to remove those to get to the air filter. First up, remove this plastic cover. It uses 3 plastic fasteners.

The battery box top has to be removed to gain access to the intake. It has 3 plastic clips near the front that hold it in place. Squeeze them together to release the lid.

This tube is directly over the air filter box. Now that the battery box lid is out of the way, you can remove it. Lift up the rear and pull it towards the back of the car.

The air filter box is now visible. There are 4 metal clamps, one at each corner.

Unclamp each of the air box clamps.

Open the intake box and remove the filter.

Inspect the air filter for dirt. This one isn't terrible yet, but it won't hurt to replace it now.

Place your new filter in the intake box, with the flatter side facing up.

Carefully position the intake box lid over the filter (mine wanted to grab onto the filter's rubber gasket near the engine). To make it easier, I lifted slightly on the engine side of the box, positioned the lid, then closed the box lid in place.

Re-clamp the 4 metal clamps at each corner. If they are too difficult to close, then your air filter is not installed correctly or the box lid is not lined up. Reposition and try again.

Slide the front of the battery vent tube in place. Note that it has a slot in it to for alignment. Place the back edge on the lip of the battery box.

Slide the rear tabs of the battery box into place, and clamp the battery box down in front by pressing it closed. The vent tube should be trapped in place by the box.

Replace the plastic trim panel and re-install all 3 of the plastic fasteners.