Find the volume of the solid bounded by the xy plane, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=4$.
$begingroup$
Find the volume of the solid bounded by the xy plane, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=4$.
If we draw the graph, then the integral will be calculated should be
$$ int_{0}^{2} int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx $$
This would give the volume of the solid in 1st quadrant which can also be obtained through
$$ int_{0}^{2} int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy $$
which would be equal to $4pi - frac83$. Both the above equations give the same result.
But if we try to find the volume of the entire solid formed by the three curves, then the results from the above equations(after changing the limits appropriately) don't match.
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx = 2(8pi-frac{16}{3})$$
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy = 16pi$$
why the results of the these two equations don't match? Is there some problem with the limits I've set or something else? Please help
PS: There is a similar question, but my query is different.
multiple-integral
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
$endgroup$
add a comment |
$begingroup$
Find the volume of the solid bounded by the xy plane, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=4$.
If we draw the graph, then the integral will be calculated should be
$$ int_{0}^{2} int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx $$
This would give the volume of the solid in 1st quadrant which can also be obtained through
$$ int_{0}^{2} int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy $$
which would be equal to $4pi - frac83$. Both the above equations give the same result.
But if we try to find the volume of the entire solid formed by the three curves, then the results from the above equations(after changing the limits appropriately) don't match.
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx = 2(8pi-frac{16}{3})$$
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy = 16pi$$
why the results of the these two equations don't match? Is there some problem with the limits I've set or something else? Please help
PS: There is a similar question, but my query is different.
multiple-integral
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
$endgroup$
add a comment |
$begingroup$
Find the volume of the solid bounded by the xy plane, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=4$.
If we draw the graph, then the integral will be calculated should be
$$ int_{0}^{2} int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx $$
This would give the volume of the solid in 1st quadrant which can also be obtained through
$$ int_{0}^{2} int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy $$
which would be equal to $4pi - frac83$. Both the above equations give the same result.
But if we try to find the volume of the entire solid formed by the three curves, then the results from the above equations(after changing the limits appropriately) don't match.
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx = 2(8pi-frac{16}{3})$$
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy = 16pi$$
why the results of the these two equations don't match? Is there some problem with the limits I've set or something else? Please help
PS: There is a similar question, but my query is different.
multiple-integral
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
$endgroup$
Find the volume of the solid bounded by the xy plane, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=4$.
If we draw the graph, then the integral will be calculated should be
$$ int_{0}^{2} int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx $$
This would give the volume of the solid in 1st quadrant which can also be obtained through
$$ int_{0}^{2} int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy $$
which would be equal to $4pi - frac83$. Both the above equations give the same result.
But if we try to find the volume of the entire solid formed by the three curves, then the results from the above equations(after changing the limits appropriately) don't match.
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-x^{2}}} (4-y) : dy dx = 2(8pi-frac{16}{3})$$
$$ int_{-2}^{2} 2int_{0}^{sqrt{4-y^{2}}} (4-y) : dx dy = 16pi$$
why the results of the these two equations don't match? Is there some problem with the limits I've set or something else? Please help
PS: There is a similar question, but my query is different.
multiple-integral
multiple-integral
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
edited Dec 23 '18 at 18:14
Ajay Choudhary
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
asked Dec 23 '18 at 18:00
Ajay ChoudharyAjay Choudhary
888
888
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The limits of the second integral are wrong. It should be $$2int_0^2int_{-sqrt{4-y^{2}}}^{sqrt{4-y^{2}}} (4-y) : dx dy=4int_0^2int_0^{sqrt{4-y^{2}}} (4-y) : dx dy$$
The answers match.
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
$endgroup$
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%2f3050567%2ffind-the-volume-of-the-solid-bounded-by-the-xy-plane-the-cylinder-x2-y2%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
$begingroup$
The limits of the second integral are wrong. It should be $$2int_0^2int_{-sqrt{4-y^{2}}}^{sqrt{4-y^{2}}} (4-y) : dx dy=4int_0^2int_0^{sqrt{4-y^{2}}} (4-y) : dx dy$$
The answers match.
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
$endgroup$
add a comment |
$begingroup$
The limits of the second integral are wrong. It should be $$2int_0^2int_{-sqrt{4-y^{2}}}^{sqrt{4-y^{2}}} (4-y) : dx dy=4int_0^2int_0^{sqrt{4-y^{2}}} (4-y) : dx dy$$
The answers match.
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
$endgroup$
add a comment |
$begingroup$
The limits of the second integral are wrong. It should be $$2int_0^2int_{-sqrt{4-y^{2}}}^{sqrt{4-y^{2}}} (4-y) : dx dy=4int_0^2int_0^{sqrt{4-y^{2}}} (4-y) : dx dy$$
The answers match.
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
$endgroup$
The limits of the second integral are wrong. It should be $$2int_0^2int_{-sqrt{4-y^{2}}}^{sqrt{4-y^{2}}} (4-y) : dx dy=4int_0^2int_0^{sqrt{4-y^{2}}} (4-y) : dx dy$$
The answers match.
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
edited Dec 23 '18 at 18:19
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
answered Dec 23 '18 at 18:06
Shubham JohriShubham Johri
5,475818
5,475818
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%2f3050567%2ffind-the-volume-of-the-solid-bounded-by-the-xy-plane-the-cylinder-x2-y2%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
