Divergent Nozzle Homework: Integrating to Get 5

[TimeRip496]

Homework Statement

where the boldface type denotes a vector quantity. In general, the incremental surface dS may be expressed as r dψ R d∅. Examining Figure 1, we see that dS = R² sin∅ dψ d∅ for this particular nozzle. Substituting Equations 3 and 4 in Equation 2 and integrating ∅ from zero to θ and ψ circumferentially from zero to π, we arrive at the result

Homework Equations

The Attempt at a Solution

When I integrate to get 5, I get 0.5ρv_e²R²sin²θ instead. Shouldn't there be a 0.5 when you integrate sinθcosθ or did the paper miss out on something?

Source: https://mdao.grc.nasa.gov/publications/NASA-TM-105176-Diverge.pdf

 

[KataKoniK]

It is possible that there is a typo in the paper and the 0.5 is missing. However, it is also possible that the 0.5 was intentionally left out because it cancels out when the equation is simplified. Without more context or information, it is difficult to determine the reason for the discrepancy. It may be helpful to double check your integration and also consult other sources for the correct equation.