How to increase the air intake

Brass instruments work with air. There's no  way around it. Use your air well and you will play well. The more air you can take, the easier it gets to make a nice tone, play in the high register and have more endurance. Of course it doesn't all comes down to how much air you can take in your lungs but air intake does play a major role in your playing. 

First you need to follow the laws of physics: think of your body as a balloon; when it gets bigger, air goes in. When it gets smaller, air goes out. To increase your air intake, you'll have to facilitate this movement. 

To do so you will have to get rid of  unnecessary tensions in your body:

-Don't press on your valves too hard. It will create stress in your harm which will affect your torso and disrupt the natural movement of your diaphragm and rib cage.

-Don't force with your forehead. Tensions in your upper facial muscles won't help you reaching high notes on the contrary. 

-Keep your shoulders down. Raising your shoulders while playing will hurt you on the short term as well as on the long term. 

-Don't hold your instrument too tight. 

-Don't keep your harms to close to your body. This is a defensive reflex that most people have to protect themselves against danger. Unfortunately, this will not help you here. To counter this habit, you can slightly raise your elbows for each breath you take. You don't need to lift them to the roof. Just a bit to make sure your rib cage has enough space to expand. Eventually, you will get the right habit and won't need to think about it any more. 

-Take  Alexander technique lessons. In a previous post, I mentioned how great my playing felt after each Alexander lesson. A good Alexander technique teacher will help you use your body in a way that will make your breathing natural and fluid. I strongly recommend this to all musicians. 

-Do breathing exercises. For example: 
Stand up and open your harms while taking deep breaths. 
Blow on your hand while moving it up and down. 
Lie down on your back and feel your body inflating and deflating on each breath.

-Be in good physical shape.  

Increasing you air intake will take some time but if you do these simple things you should see some result immediately. Keep working on it and you'll develop a nice fluid air flow to support your playing.