# Where does this integral for a 'aerospike nozzle' come from?

Today I was reading about the aerospike nozzle and found this image:

(from section 10.3.1 of this lecture)

This image tells us how to measure the thrust of this kind of nozzle. I can understand all of it except this part:

Where does this integral come from? How do they derive it?

• Where is the image from? Besides citing your source being a common courtesy, it might make answering the question easier. – Jan Hudec May 1 '17 at 16:06
• @JanHudec here is a link to the source of the image: web.csulb.edu/colleges/coe/ae/engr370i/ch10/sect_3-1 – Roh May 1 '17 at 16:31
• On this site question (and answers) are editable and additional information should be edited in. I've added the citation for you now. – Jan Hudec May 1 '17 at 16:44

Now turn the conventional nozzle inside out. The expansion happens on its outside, but there is still a pressure acting on a backward-facing surface. The graph doesn't explain it, but the ramp area $\text{A}_{\text{Ramp}}$ is the projection of the nozzle area in flight direction. The ramp pressure $\text{p}_{\text{Ramp}}$ is the pressure acting along the nozzle surface, and it must be reduced by ambient pressure $\text{p}_{\infty}$ in order to arrive at the correct thrust. The integration is needed because $\text{p}_{\text{Ramp}}$ varies over the nozzle area.