Definite Integration Scaling
$begingroup$
Probably a very easy question, sorry.
Suppose we fix some $lambda in mathbb{R}$ and some integrable function $f : [a,b] to mathbb{R}$.
Consider $f_lambda(x) = f(lambda x)$. I want to show that $f(lambda x)$ is integrable of $[tfrac{a}{lambda},tfrac{b}{lambda}]$, and:
$$int_{(tfrac{a}{lambda})}^{(tfrac{b}{lambda})} f_lambda =
frac{1}{lambda}int_a^b f$$
How would I go about showing this from partitions?
(I have shown multiplication by a constant:
$int_a^b lambda f = lambda int_a^b f$, and I tried the same approach. Would I just show it through summations again? I am not quite sure how this translates to an interval and its partition).
real-analysis integration definite-integrals
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
$endgroup$
add a comment |
$begingroup$
Probably a very easy question, sorry.
Suppose we fix some $lambda in mathbb{R}$ and some integrable function $f : [a,b] to mathbb{R}$.
Consider $f_lambda(x) = f(lambda x)$. I want to show that $f(lambda x)$ is integrable of $[tfrac{a}{lambda},tfrac{b}{lambda}]$, and:
$$int_{(tfrac{a}{lambda})}^{(tfrac{b}{lambda})} f_lambda =
frac{1}{lambda}int_a^b f$$
How would I go about showing this from partitions?
(I have shown multiplication by a constant:
$int_a^b lambda f = lambda int_a^b f$, and I tried the same approach. Would I just show it through summations again? I am not quite sure how this translates to an interval and its partition).
real-analysis integration definite-integrals
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
$endgroup$
add a comment |
$begingroup$
Probably a very easy question, sorry.
Suppose we fix some $lambda in mathbb{R}$ and some integrable function $f : [a,b] to mathbb{R}$.
Consider $f_lambda(x) = f(lambda x)$. I want to show that $f(lambda x)$ is integrable of $[tfrac{a}{lambda},tfrac{b}{lambda}]$, and:
$$int_{(tfrac{a}{lambda})}^{(tfrac{b}{lambda})} f_lambda =
frac{1}{lambda}int_a^b f$$
How would I go about showing this from partitions?
(I have shown multiplication by a constant:
$int_a^b lambda f = lambda int_a^b f$, and I tried the same approach. Would I just show it through summations again? I am not quite sure how this translates to an interval and its partition).
real-analysis integration definite-integrals
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
$endgroup$
Probably a very easy question, sorry.
Suppose we fix some $lambda in mathbb{R}$ and some integrable function $f : [a,b] to mathbb{R}$.
Consider $f_lambda(x) = f(lambda x)$. I want to show that $f(lambda x)$ is integrable of $[tfrac{a}{lambda},tfrac{b}{lambda}]$, and:
$$int_{(tfrac{a}{lambda})}^{(tfrac{b}{lambda})} f_lambda =
frac{1}{lambda}int_a^b f$$
How would I go about showing this from partitions?
(I have shown multiplication by a constant:
$int_a^b lambda f = lambda int_a^b f$, and I tried the same approach. Would I just show it through summations again? I am not quite sure how this translates to an interval and its partition).
real-analysis integration definite-integrals
real-analysis integration definite-integrals
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
asked Dec 5 '18 at 14:03
Eetu KoskelaEetu Koskela
708
708
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $P={x_0,dots,x_N}$ be a partition of $[a,b]$ and $xi_kin[x_{k-1},x_k]$. Then
begin{align}
sum_{k=1}^Nf(xi_k)(x_k-x_{k-1})&=lambdasum_{k=1}^NfBigl(lambda,frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr)\
&=lambdasum_{k=1}^Nf_lambdaBigl(frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr).
end{align}The first summation is a Riemann sum for $int_a^bf$, and the last one for $int_{a/lambda}^{b/lambda}f_lambda$.
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027102%2fdefinite-integration-scaling%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $P={x_0,dots,x_N}$ be a partition of $[a,b]$ and $xi_kin[x_{k-1},x_k]$. Then
begin{align}
sum_{k=1}^Nf(xi_k)(x_k-x_{k-1})&=lambdasum_{k=1}^NfBigl(lambda,frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr)\
&=lambdasum_{k=1}^Nf_lambdaBigl(frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr).
end{align}The first summation is a Riemann sum for $int_a^bf$, and the last one for $int_{a/lambda}^{b/lambda}f_lambda$.
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
add a comment |
$begingroup$
Let $P={x_0,dots,x_N}$ be a partition of $[a,b]$ and $xi_kin[x_{k-1},x_k]$. Then
begin{align}
sum_{k=1}^Nf(xi_k)(x_k-x_{k-1})&=lambdasum_{k=1}^NfBigl(lambda,frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr)\
&=lambdasum_{k=1}^Nf_lambdaBigl(frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr).
end{align}The first summation is a Riemann sum for $int_a^bf$, and the last one for $int_{a/lambda}^{b/lambda}f_lambda$.
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
add a comment |
$begingroup$
Let $P={x_0,dots,x_N}$ be a partition of $[a,b]$ and $xi_kin[x_{k-1},x_k]$. Then
begin{align}
sum_{k=1}^Nf(xi_k)(x_k-x_{k-1})&=lambdasum_{k=1}^NfBigl(lambda,frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr)\
&=lambdasum_{k=1}^Nf_lambdaBigl(frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr).
end{align}The first summation is a Riemann sum for $int_a^bf$, and the last one for $int_{a/lambda}^{b/lambda}f_lambda$.
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
$endgroup$
Let $P={x_0,dots,x_N}$ be a partition of $[a,b]$ and $xi_kin[x_{k-1},x_k]$. Then
begin{align}
sum_{k=1}^Nf(xi_k)(x_k-x_{k-1})&=lambdasum_{k=1}^NfBigl(lambda,frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr)\
&=lambdasum_{k=1}^Nf_lambdaBigl(frac{xi_k}{lambda}Bigr)Bigl(frac{x_k}{lambda}-frac{x_{k-1}}{lambda}Bigr).
end{align}The first summation is a Riemann sum for $int_a^bf$, and the last one for $int_{a/lambda}^{b/lambda}f_lambda$.
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
answered Dec 5 '18 at 15:00
Julián AguirreJulián Aguirre
68.1k24094
68.1k24094
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027102%2fdefinite-integration-scaling%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
