Mixing
about exchange the order of double integral
How can I exchange the order of the double integral $$int_{0}^{2pi}int_0^{sin x}f(x,y)dydx$$
My attempt : I think maybe I can use the function arcsinx but the function’s domain is
[-1,1] I think maybe I should translation and rotation the function
calculus integration analysis
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
asked Nov 22 '18 at 0:42
jacksonjackson
798
add a comment |
How can I exchange the order of the double integral $$int_{0}^{2pi}int_0^{sin x}f(x,y)dydx$$
My attempt : I think maybe I can use the function arcsinx but the function’s domain is
[-1,1] I think maybe I should translation and rotation the function
calculus integration analysis
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
asked Nov 22 '18 at 0:42
jacksonjackson
798
Here you can exchange it freely while you sure that the function is integrable. (As none of the bounds of the integral depend on one of the variables.)
– kolobokish
Nov 22 '18 at 0:57
Aaaaaaa sorry I tap wrong ...
– jackson
Nov 22 '18 at 0:58
add a comment |
How can I exchange the order of the double integral $$int_{0}^{2pi}int_0^{sin x}f(x,y)dydx$$
My attempt : I think maybe I can use the function arcsinx but the function’s domain is
[-1,1] I think maybe I should translation and rotation the function
calculus integration analysis
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
asked Nov 22 '18 at 0:42
jacksonjackson
798
How can I exchange the order of the double integral $$int_{0}^{2pi}int_0^{sin x}f(x,y)dydx$$
My attempt : I think maybe I can use the function arcsinx but the function’s domain is
[-1,1] I think maybe I should translation and rotation the function
calculus integration analysis
calculus integration analysis
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
asked Nov 22 '18 at 0:42
jacksonjackson
798
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
asked Nov 22 '18 at 0:42
jacksonjackson
798
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
edited Nov 22 '18 at 1:08
Graham Kemp
84.7k43378
84.7k43378
asked Nov 22 '18 at 0:42
jacksonjackson
798
asked Nov 22 '18 at 0:42
jacksonjackson
798
asked Nov 22 '18 at 0:42
jacksonjackson
798
798
Here you can exchange it freely while you sure that the function is integrable. (As none of the bounds of the integral depend on one of the variables.)
– kolobokish
Nov 22 '18 at 0:57
Aaaaaaa sorry I tap wrong ...
– jackson
Nov 22 '18 at 0:58
add a comment |
Here you can exchange it freely while you sure that the function is integrable. (As none of the bounds of the integral depend on one of the variables.)
– kolobokish
Nov 22 '18 at 0:57
Aaaaaaa sorry I tap wrong ...
– jackson
Nov 22 '18 at 0:58
Here you can exchange it freely while you sure that the function is integrable. (As none of the bounds of the integral depend on one of the variables.)
– kolobokish
Nov 22 '18 at 0:57
Here you can exchange it freely while you sure that the function is integrable. (As none of the bounds of the integral depend on one of the variables.)
– kolobokish
Nov 22 '18 at 0:57
Aaaaaaa sorry I tap wrong ...
– jackson
Nov 22 '18 at 0:58
Aaaaaaa sorry I tap wrong ...
– jackson
Nov 22 '18 at 0:58
add a comment |
2 Answers
2
active
oldest
votes
Partition the interval for $x$ into $[0,pi/2), [pi/2,3pi/2) $ and $[3pi/2,2pi)$ giving you a sum of three integrals.
Exchange the order of integration for each of these using three versions of $arcsin$ that map to the appropriate Interval.
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
Can you teach me what should I do such like [$pi$/2,3$pi$/2]
– jackson
Nov 22 '18 at 1:26
add a comment |
You can use Fubini's theorem
$$
color{blue}{int_{0}^{2pi}{rm d}x} color{red}{int_{0}^{pi/2}{rm d}y} ~ f(x,y) = color{red}{int_{0}^{pi/2}{rm d}y} color{blue}{int_{0}^{2pi}{rm d}x} ~ f(x,y)
$$
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
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%2f3008601%2fabout-exchange-the-order-of-double-integral%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Partition the interval for $x$ into $[0,pi/2), [pi/2,3pi/2) $ and $[3pi/2,2pi)$ giving you a sum of three integrals.
Exchange the order of integration for each of these using three versions of $arcsin$ that map to the appropriate Interval.
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
Can you teach me what should I do such like [$pi$/2,3$pi$/2]
– jackson
Nov 22 '18 at 1:26
add a comment |
Partition the interval for $x$ into $[0,pi/2), [pi/2,3pi/2) $ and $[3pi/2,2pi)$ giving you a sum of three integrals.
Exchange the order of integration for each of these using three versions of $arcsin$ that map to the appropriate Interval.
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
Can you teach me what should I do such like [$pi$/2,3$pi$/2]
– jackson
Nov 22 '18 at 1:26
add a comment |
Partition the interval for $x$ into $[0,pi/2), [pi/2,3pi/2) $ and $[3pi/2,2pi)$ giving you a sum of three integrals.
Exchange the order of integration for each of these using three versions of $arcsin$ that map to the appropriate Interval.
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
Partition the interval for $x$ into $[0,pi/2), [pi/2,3pi/2) $ and $[3pi/2,2pi)$ giving you a sum of three integrals.
Exchange the order of integration for each of these using three versions of $arcsin$ that map to the appropriate Interval.
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
answered Nov 22 '18 at 1:16
Graham KempGraham Kemp
84.7k43378
84.7k43378
Can you teach me what should I do such like [$pi$/2,3$pi$/2]
– jackson
Nov 22 '18 at 1:26
add a comment |
Can you teach me what should I do such like [$pi$/2,3$pi$/2]
– jackson
Nov 22 '18 at 1:26
Can you teach me what should I do such like [$pi$/2,3$pi$/2]
– jackson
Nov 22 '18 at 1:26
Can you teach me what should I do such like [$pi$/2,3$pi$/2]
– jackson
Nov 22 '18 at 1:26
add a comment |
You can use Fubini's theorem
$$
color{blue}{int_{0}^{2pi}{rm d}x} color{red}{int_{0}^{pi/2}{rm d}y} ~ f(x,y) = color{red}{int_{0}^{pi/2}{rm d}y} color{blue}{int_{0}^{2pi}{rm d}x} ~ f(x,y)
$$
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
add a comment |
You can use Fubini's theorem
$$
color{blue}{int_{0}^{2pi}{rm d}x} color{red}{int_{0}^{pi/2}{rm d}y} ~ f(x,y) = color{red}{int_{0}^{pi/2}{rm d}y} color{blue}{int_{0}^{2pi}{rm d}x} ~ f(x,y)
$$
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
add a comment |
You can use Fubini's theorem
$$
color{blue}{int_{0}^{2pi}{rm d}x} color{red}{int_{0}^{pi/2}{rm d}y} ~ f(x,y) = color{red}{int_{0}^{pi/2}{rm d}y} color{blue}{int_{0}^{2pi}{rm d}x} ~ f(x,y)
$$
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
You can use Fubini's theorem
$$
color{blue}{int_{0}^{2pi}{rm d}x} color{red}{int_{0}^{pi/2}{rm d}y} ~ f(x,y) = color{red}{int_{0}^{pi/2}{rm d}y} color{blue}{int_{0}^{2pi}{rm d}x} ~ f(x,y)
$$
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
answered Nov 22 '18 at 0:57
caveraccaverac
14k21130
14k21130
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3008601%2fabout-exchange-the-order-of-double-integral%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
Here you can exchange it freely while you sure that the function is integrable. (As none of the bounds of the integral depend on one of the variables.)
– kolobokish
Nov 22 '18 at 0:57
Aaaaaaa sorry I tap wrong ...
– jackson
Nov 22 '18 at 0:58