How is τmax/rmax related to the second moment of area?

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The discussion centers on the relationship between τmax/rmax and the second moment of area, specifically how to derive the equation for T using integration. The user is confused about the integration process and questions how their professor arrived at the answer involving the second moment of area, I = ∫A r^2 dA. They note that integrating r^2 would yield r^3/3 + C, indicating a misunderstanding of the integration limits or context. Clarification on the integration process and its application to the second moment of area is sought. Understanding this relationship is crucial for solving related problems in mechanics.
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Homework Statement



dT= τmax/rmax * r^2 dA

T = τmax/rmax ∫▒〖r^2 dA〗

as far as I am concerned integrating this will give us r^3/3 + c?
or am i thinking wrong, how did my teacher get the answer below in bold

Homework Equations


its not a great deal to do but I don't know how my professor got the following answer below


The Attempt at a Solution


the question is why the answer to this is

τ/rmax * Ip
 
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The second moment of area is defined as

I = \int_A r^2 dA


had you ∫r2 dr, then you'd get r3/3 + C
 

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