Integration: Solve Nm sin2α = ωR * intgrl α:β [(cosβ − cosθ)sinθ cosθ dθ]

[palm2694]

Integration: Solve Nm sin2α = ωR * intgrl α:β [(cosβ − cosθ)sinθ cosθ dθ]

Hi,

Could anyone help me with the steps to solving this integral:

N_m sin²α = ωR * intgrl α:β [(cosβ − cosθ)sinθ cosθ dθ]

Solution:

N_m =ωR/sin²α*[1/3(cos³α − cos³β) −cosβ/4(cos 2α − cos 2β)]

See image below:

(I'm basically req'd to integrate the over the region of b):

 

[Ishuda]

Could anyone help me with the steps to solving this integral:

N_m sin²α = ωR * intgrl α:β [(cosβ − cosθ)sinθ cosθ dθ]

Solution:

N_m =ωR/sin²α*[1/3(cos³α − cos³β) −cosβ/4(cos 2α − cos 2β)]

See image below:

(I'm basically req'd to integrate the over the region of b):

View attachment 8865

Start with

\(\displaystyle \displaystyle \int_\alpha^\beta\, [cos(\beta)\, - cos(\theta)]\, sin(\theta)\, cos(\theta)\,d\theta\)

. . .\(\displaystyle \displaystyle =\, cos(\beta)\,\int_\alpha^\beta\, sin(\theta)\, cos(\theta)\,d\theta\, -\, \int_\alpha^\beta\, sin(\theta)\, cos^2(\theta)\,d\theta\)

You should know the solution to the two integrals. HINT:

\(\displaystyle \dfrac{d\,cos(\theta)}{d\theta}\, =\, -\,sin(\theta)\)