Computing Riemannian integrals from paritions.
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I need help understanding how does one show definite Riemannian integration from $a$ to $b$ directly from sums if $a,b$ are not given. The text I am working from talks about odd and even functions and recommends trying on $f(x) = x^2$ and $f(x) = x^3$ on any closed interval $[a,b]$.
I know I need to arrive at $lim U(f,P_n) = lim L(f,P_n)$, but I am not sure how to show that properly with careful rigorous proofs how to do that. Moreover, it seems that text implies that I need to use even/odd properties of the functions somehow, which confuses me. The text only has an example for linear case, and non-rigorous example showing how Riemannian integration works at all in principle. (Without using Fundamental Theorem).
I would really appriciate a demonstration of how to do that in principle on these two functions above. Thank you in advance.
real-analysis calculus integration definite-integrals
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
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add a comment |
$begingroup$
I need help understanding how does one show definite Riemannian integration from $a$ to $b$ directly from sums if $a,b$ are not given. The text I am working from talks about odd and even functions and recommends trying on $f(x) = x^2$ and $f(x) = x^3$ on any closed interval $[a,b]$.
I know I need to arrive at $lim U(f,P_n) = lim L(f,P_n)$, but I am not sure how to show that properly with careful rigorous proofs how to do that. Moreover, it seems that text implies that I need to use even/odd properties of the functions somehow, which confuses me. The text only has an example for linear case, and non-rigorous example showing how Riemannian integration works at all in principle. (Without using Fundamental Theorem).
I would really appriciate a demonstration of how to do that in principle on these two functions above. Thank you in advance.
real-analysis calculus integration definite-integrals
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
$endgroup$
$begingroup$
Try exactly the same approach as used for linear case and let us know where you are facing the problem.
$endgroup$
– Paramanand Singh
Dec 5 '18 at 17:14
add a comment |
$begingroup$
I need help understanding how does one show definite Riemannian integration from $a$ to $b$ directly from sums if $a,b$ are not given. The text I am working from talks about odd and even functions and recommends trying on $f(x) = x^2$ and $f(x) = x^3$ on any closed interval $[a,b]$.
I know I need to arrive at $lim U(f,P_n) = lim L(f,P_n)$, but I am not sure how to show that properly with careful rigorous proofs how to do that. Moreover, it seems that text implies that I need to use even/odd properties of the functions somehow, which confuses me. The text only has an example for linear case, and non-rigorous example showing how Riemannian integration works at all in principle. (Without using Fundamental Theorem).
I would really appriciate a demonstration of how to do that in principle on these two functions above. Thank you in advance.
real-analysis calculus integration definite-integrals
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
$endgroup$
I need help understanding how does one show definite Riemannian integration from $a$ to $b$ directly from sums if $a,b$ are not given. The text I am working from talks about odd and even functions and recommends trying on $f(x) = x^2$ and $f(x) = x^3$ on any closed interval $[a,b]$.
I know I need to arrive at $lim U(f,P_n) = lim L(f,P_n)$, but I am not sure how to show that properly with careful rigorous proofs how to do that. Moreover, it seems that text implies that I need to use even/odd properties of the functions somehow, which confuses me. The text only has an example for linear case, and non-rigorous example showing how Riemannian integration works at all in principle. (Without using Fundamental Theorem).
I would really appriciate a demonstration of how to do that in principle on these two functions above. Thank you in advance.
real-analysis calculus integration definite-integrals
real-analysis calculus integration definite-integrals
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
edited Dec 5 '18 at 16:55
Eetu Koskela
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
asked Dec 5 '18 at 16:04
Eetu KoskelaEetu Koskela
708
708
$begingroup$
Try exactly the same approach as used for linear case and let us know where you are facing the problem.
$endgroup$
– Paramanand Singh
Dec 5 '18 at 17:14
add a comment |
$begingroup$
Try exactly the same approach as used for linear case and let us know where you are facing the problem.
$endgroup$
– Paramanand Singh
Dec 5 '18 at 17:14
$begingroup$
Try exactly the same approach as used for linear case and let us know where you are facing the problem.
$endgroup$
– Paramanand Singh
Dec 5 '18 at 17:14
$begingroup$
Try exactly the same approach as used for linear case and let us know where you are facing the problem.
$endgroup$
– Paramanand Singh
Dec 5 '18 at 17:14
add a comment |
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$begingroup$
Try exactly the same approach as used for linear case and let us know where you are facing the problem.
$endgroup$
– Paramanand Singh
Dec 5 '18 at 17:14