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enter image description here

In the above image we can see to find area of circle they consider the very small annulus with area of $\delta A=2\pi r \delta r$. and in order to to find $\int u^3dA$ and it plugged in equation of $u_r$ into integral.

My question is : We consider the $\delta A$ as area of annulus and find the integral with bound $r\in [0,R]$. which give the area of circle. I wonder what we call this small area $\delta A$ in math? Is this riemenn?

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    $\begingroup$ $\delta A$ is the infinitesimal area enclosed between circles of radii $r$ and $r+\delta r$. Instead of differentiating, you might as well find it by $\delta A = \pi (r+\delta r)^2 - \pi r^2 = \pi (\delta r)^2 + 2\pi r\delta r \approx 2\pi r\delta r$. Here, we choose to neglect the higher order (second and larger) terms. $\endgroup$ – strawberry-sunshine Nov 22 at 8:45
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    $\begingroup$ Also, yes, this is a Riemann sum. $\endgroup$ – strawberry-sunshine Nov 22 at 8:46
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    $\begingroup$ @strawberry-sunshine . I understood. thank you very much for making it clear to me. $\endgroup$ – soheil Nov 22 at 8:50
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That kind of quantity is known as a differential.

Basically, a differential of a quantity-A dependent on another quantity-B is the amount of increase of quantity-A as we nudge a little of quantity B i.e: a relation between increments.

This video discusses it see here

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