This is a follow up from another post . I was using the integration symbol available in the Basic Math Assistance available in Wolfram Mathematica.

I am new to vector calculus operations. There is a known identity found in the textbooks

$$\int _{4 \pi }\overset{\rightharpoonup }{s} (\overset{\rightharpoonup }{s}\cdot A) d \omega=\frac{1}{3} (4 \pi ) A$$

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[\[Theta]], 
   Cos[\[Theta]]}*(Dot[{Sin[\[Theta]], Cos[\[Theta]]}, {a1, 
     a2}]), \[Theta], {0, 4 \[Pi]}]

It is obviosu that I am doing something fundamentally not correct. I go to WM documentation on Vector Calculus but does not offer much substance or examples. How will you enter the equation above in order to return the identity in the right?