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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 $\frac{d}{da}f(x)$?

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Use the fact that

$$\int_{0}^{a}{x^2dx}=\frac{1}{3}a^3$$

More generally,

$$\frac{d}{dt}\int_{a(t)}^{b(t)}{f(x)dx}=\frac{d}{dt}(F(b(t))-F(a(t)))=b'(t)f(b(t))-a'(t)f(a(t))$$

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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