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The rate at which the value of a definite integral increases
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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 $fracddaf(x)$?
calculus integration
edited Sep 9 at 6:23
dmtri
850317
asked Sep 9 at 6:16
Mitchell Browne
185311
add a comment |Â
up vote
5
down vote
favorite
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 $fracddaf(x)$?
calculus integration
edited Sep 9 at 6:23
dmtri
850317
asked Sep 9 at 6:16
Mitchell Browne
185311
math.stackexchange.com/questions/1261432/…
– Ahmed S. Attaalla
Sep 9 at 6:44
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
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 $fracddaf(x)$?
calculus integration
edited Sep 9 at 6:23
dmtri
850317
asked Sep 9 at 6:16
Mitchell Browne
185311
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 $fracddaf(x)$?
calculus integration
calculus integration
edited Sep 9 at 6:23
dmtri
850317
asked Sep 9 at 6:16
Mitchell Browne
185311
edited Sep 9 at 6:23
dmtri
850317
asked Sep 9 at 6:16
Mitchell Browne
185311
edited Sep 9 at 6:23
dmtri
850317
edited Sep 9 at 6:23
dmtri
850317
edited Sep 9 at 6:23
dmtri
850317
850317
asked Sep 9 at 6:16
Mitchell Browne
185311
asked Sep 9 at 6:16
Mitchell Browne
185311
asked Sep 9 at 6:16
Mitchell Browne
185311
185311
math.stackexchange.com/questions/1261432/…
– Ahmed S. Attaalla
Sep 9 at 6:44
add a comment |Â
math.stackexchange.com/questions/1261432/…
– Ahmed S. Attaalla
Sep 9 at 6:44
math.stackexchange.com/questions/1261432/…
– Ahmed S. Attaalla
Sep 9 at 6:44
math.stackexchange.com/questions/1261432/…
– Ahmed S. Attaalla
Sep 9 at 6:44
add a comment |Â
1 Answer
1
active
oldest
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up vote
8
down vote
accepted
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 $t$.
answered Sep 9 at 6:18
stressed out
3,6101431
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
8
down vote
accepted
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 $t$.
answered Sep 9 at 6:18
stressed out
3,6101431
add a comment |Â
up vote
8
down vote
accepted
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 $t$.
answered Sep 9 at 6:18
stressed out
3,6101431
add a comment |Â
up vote
8
down vote
accepted
up vote
8
down vote
accepted
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 $t$.
answered Sep 9 at 6:18
stressed out
3,6101431
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 $t$.
answered Sep 9 at 6:18
stressed out
3,6101431
answered Sep 9 at 6:18
stressed out
3,6101431
answered Sep 9 at 6:18
stressed out
3,6101431
answered Sep 9 at 6:18
stressed out
3,6101431
3,6101431
add a comment |Â
add a comment |Â
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math.stackexchange.com/questions/1261432/…
– Ahmed S. Attaalla
Sep 9 at 6:44