This is a follow up from another [post][1]. I was using the integration symbols available in the Basic Math Assistant palette. 

I am new to vector calculus operations. There is a known identity found in my textbook.

$\qquad \int _{4 \pi }\hat{s}   (\hat{s}\cdot A) d \omega=\frac{4 \pi}{3}A$

I have no idea how to do this type of integration. This is what I tried but return a dissaster

    Integrate[s*(Dot[s, A]), s, {0, 4 \[Pi]}]

Also, without success

    Integrate[{Sin[θ], Cos[θ]}*(Dot[{Sin[θ], Cos[θ]}, {a1, a2}]), θ, {0, 4 π}]

It is obvious that I am doing something fundamentally not correct. I go to the documentation on Vector Calculus, but it does not offer much in substance or examples. How do you enter the integral expression shown above in order to return the identity in the right?

###Update

In response to comments, here is a copy of the text.  This is from page 10 *Optical-Thermal Response of Laser-Irradiated Tissue* ISBN 9789048188307

$\omega$ is the surface area of a sphere in solid angle steradians. $\hat s$ is the directional vector of a pencil of radiation located inside the sphere

[![enter image description here][2]][2]


  [1]: https://mathematica.stackexchange.com/questions/194305/latex-format-copy-problem
  [2]: https://i.sstatic.net/tpie9.jpg