Mixing
Double integral change of variables to polar coordinates.
$begingroup$
Sketch the region represented in the following double integral:
$I =∬ln(x^2+y^2)dxdy$
In the region:
$x:ycotβ$ to $sqrt(a^2+y^2)$
$y:0$ to $asinβ$
where a > 0 and 0 < β < π/2.
By changing to polar coordinates or otherwise, show that
I = $a^2β(ln a − 1/2)$.
So far I have found that the sketch yields a triangle inscribed in the circle of radius $a$ in the first quadrant. I am having trouble converting to polar coordinates.
calculus integration multivariable-calculus
asked Feb 3 at 11:29
matmat
12
$endgroup$
add a comment |
$begingroup$
Sketch the region represented in the following double integral:
$I =∬ln(x^2+y^2)dxdy$
In the region:
$x:ycotβ$ to $sqrt(a^2+y^2)$
$y:0$ to $asinβ$
where a > 0 and 0 < β < π/2.
By changing to polar coordinates or otherwise, show that
I = $a^2β(ln a − 1/2)$.
So far I have found that the sketch yields a triangle inscribed in the circle of radius $a$ in the first quadrant. I am having trouble converting to polar coordinates.
calculus integration multivariable-calculus
asked Feb 3 at 11:29
matmat
12
$endgroup$
add a comment |
$begingroup$
Sketch the region represented in the following double integral:
$I =∬ln(x^2+y^2)dxdy$
In the region:
$x:ycotβ$ to $sqrt(a^2+y^2)$
$y:0$ to $asinβ$
where a > 0 and 0 < β < π/2.
By changing to polar coordinates or otherwise, show that
I = $a^2β(ln a − 1/2)$.
So far I have found that the sketch yields a triangle inscribed in the circle of radius $a$ in the first quadrant. I am having trouble converting to polar coordinates.
calculus integration multivariable-calculus
asked Feb 3 at 11:29
matmat
12
$endgroup$
Sketch the region represented in the following double integral:
$I =∬ln(x^2+y^2)dxdy$
In the region:
$x:ycotβ$ to $sqrt(a^2+y^2)$
$y:0$ to $asinβ$
where a > 0 and 0 < β < π/2.
By changing to polar coordinates or otherwise, show that
I = $a^2β(ln a − 1/2)$.
So far I have found that the sketch yields a triangle inscribed in the circle of radius $a$ in the first quadrant. I am having trouble converting to polar coordinates.
calculus integration multivariable-calculus
calculus integration multivariable-calculus
asked Feb 3 at 11:29
matmat
12
asked Feb 3 at 11:29
matmat
12
asked Feb 3 at 11:29
matmat
12
asked Feb 3 at 11:29
matmat
12
asked Feb 3 at 11:29
matmat
12
12
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: In the polar coordinate system $x^2+y^2 Rightarrow r^2$ and $dxdy Rightarrow rcdot drdtheta$. Also, we have that $y=rcdot sin{theta}$ and $x=rcdot cos{theta}$.
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
$endgroup$
$begingroup$
My problem is finding the limits for $r$ and theta.
$endgroup$
– mat
Feb 3 at 12:09
add a comment |
Your Answer
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%2f3098466%2fdouble-integral-change-of-variables-to-polar-coordinates%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
$begingroup$
Hint: In the polar coordinate system $x^2+y^2 Rightarrow r^2$ and $dxdy Rightarrow rcdot drdtheta$. Also, we have that $y=rcdot sin{theta}$ and $x=rcdot cos{theta}$.
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
$endgroup$
$begingroup$
My problem is finding the limits for $r$ and theta.
$endgroup$
– mat
Feb 3 at 12:09
add a comment |
$begingroup$
Hint: In the polar coordinate system $x^2+y^2 Rightarrow r^2$ and $dxdy Rightarrow rcdot drdtheta$. Also, we have that $y=rcdot sin{theta}$ and $x=rcdot cos{theta}$.
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
$endgroup$
$begingroup$
My problem is finding the limits for $r$ and theta.
$endgroup$
– mat
Feb 3 at 12:09
add a comment |
$begingroup$
Hint: In the polar coordinate system $x^2+y^2 Rightarrow r^2$ and $dxdy Rightarrow rcdot drdtheta$. Also, we have that $y=rcdot sin{theta}$ and $x=rcdot cos{theta}$.
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
$endgroup$
Hint: In the polar coordinate system $x^2+y^2 Rightarrow r^2$ and $dxdy Rightarrow rcdot drdtheta$. Also, we have that $y=rcdot sin{theta}$ and $x=rcdot cos{theta}$.
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
answered Feb 3 at 11:59
Peter ForemanPeter Foreman
7,5371320
7,5371320
$begingroup$
My problem is finding the limits for $r$ and theta.
$endgroup$
– mat
Feb 3 at 12:09
add a comment |
$begingroup$
My problem is finding the limits for $r$ and theta.
$endgroup$
– mat
Feb 3 at 12:09
$begingroup$
My problem is finding the limits for $r$ and theta.
$endgroup$
– mat
Feb 3 at 12:09
$begingroup$
My problem is finding the limits for $r$ and theta.
$endgroup$
– mat
Feb 3 at 12:09
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%2f3098466%2fdouble-integral-change-of-variables-to-polar-coordinates%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
