Understanding how to simplify a double integral.
The textbook I am using states that
$int_{0}^{t} (int_{0}^{s} g(tau,phi(tau))dtau)ds$=$int_{0}^{t}(int_{tau}^{t} ds)g(tau,phi(tau))dtau$=$int_{0}^{t} (t-tau)g(tau,phi(tau))dtau$.
The 2nd equality sign makes complete sense to me, but I cannot see how to switch the order of integration over these more abstract regions of integration.
Thanks in advance. Hints are greatly appreciated!
calculus integration
asked Dec 8 at 22:29
Jungleshrimp
18411
add a comment |
The textbook I am using states that
$int_{0}^{t} (int_{0}^{s} g(tau,phi(tau))dtau)ds$=$int_{0}^{t}(int_{tau}^{t} ds)g(tau,phi(tau))dtau$=$int_{0}^{t} (t-tau)g(tau,phi(tau))dtau$.
The 2nd equality sign makes complete sense to me, but I cannot see how to switch the order of integration over these more abstract regions of integration.
Thanks in advance. Hints are greatly appreciated!
calculus integration
asked Dec 8 at 22:29
Jungleshrimp
18411
add a comment |
The textbook I am using states that
$int_{0}^{t} (int_{0}^{s} g(tau,phi(tau))dtau)ds$=$int_{0}^{t}(int_{tau}^{t} ds)g(tau,phi(tau))dtau$=$int_{0}^{t} (t-tau)g(tau,phi(tau))dtau$.
The 2nd equality sign makes complete sense to me, but I cannot see how to switch the order of integration over these more abstract regions of integration.
Thanks in advance. Hints are greatly appreciated!
calculus integration
asked Dec 8 at 22:29
Jungleshrimp
18411
The textbook I am using states that
$int_{0}^{t} (int_{0}^{s} g(tau,phi(tau))dtau)ds$=$int_{0}^{t}(int_{tau}^{t} ds)g(tau,phi(tau))dtau$=$int_{0}^{t} (t-tau)g(tau,phi(tau))dtau$.
The 2nd equality sign makes complete sense to me, but I cannot see how to switch the order of integration over these more abstract regions of integration.
Thanks in advance. Hints are greatly appreciated!
calculus integration
calculus integration
asked Dec 8 at 22:29
Jungleshrimp
18411
asked Dec 8 at 22:29
Jungleshrimp
18411
asked Dec 8 at 22:29
Jungleshrimp
18411
asked Dec 8 at 22:29
Jungleshrimp
18411
asked Dec 8 at 22:29
Jungleshrimp
18411
18411
add a comment |
add a comment |
active
oldest
votes
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%2f3031726%2funderstanding-how-to-simplify-a-double-integral%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3031726%2funderstanding-how-to-simplify-a-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
