Say I had a function $f(x) = x^2$ ,how could I find the rate at which $$\int_{0}^{a}{x^2dx}$$ increases for $a$, or more generally for any function.

Also, is this equivalent to $\frac{d}{da}f(x)$?


Use the fact that


More generally,


where $F(t)$ is the anti-derivative of $F$, i.e. $F'=f$. Note that you don't need to know $F$ to calculate the derivative of an integral with varying bounds with respect to the variable $t$.


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