Double integrals in polar coordinates

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The discussion centers on a misunderstanding of a step in solving double integrals in polar coordinates. A participant questions how their teacher transitioned from the expression involving rcos(θ)(rsin(θ)) to (r/2)³(2sin(θ)cos(θ)). They express confusion over the validity of this transformation, noting their own calculation leads to r⁴cos(θ)sin(θ). The conversation highlights the complexities of converting between forms in double integrals. Clarification on the steps involved in polar coordinate integration is sought.
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I was overlooking a problem that my teacher solved and i can't understand a step see took i was wondering if someone you tell me how she got from this step

Double integral rcos(o)(rsino)rto this

Double integral (r/2)^3(2sinocoso)
 
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From what you have written, i don't think it was possible for she to get from the 1st step to the second.
 
i don't see either, when i do i get to this step which i hope is correct(r²coso)(r²sino) => r^4cososino
 

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