The rate at which the value of a definite integral increases

Say I had a function $f(x) = x^2$ ,how could I find the rate at which $$int_0^ax^2dx$$ increases for $a$, or more generally for any function.

Also, is this equivalent to $fracddaf(x)$?

1 Answer
1

Use the fact that

$$int_0^ax^2dx=frac13a^3$$

More generally,

$$fracddtint_a(t)^b(t)f(x)dx=fracddt(F(b(t))-F(a(t)))=b'(t)f(b(t))-a'(t)f(a(t))$$

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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