Bernoulli Lemniscate - surface area and volume The Next CEO of Stack OverflowFind the volume of the solid obtained by rotating the region enclosed by the lines…Area of solid revolution using integration.Question on area under and between curves and volume of a solid by revolutionFind Volume of the Solid Obtained by Rotating Region(Disk Method)Cylinder volume with curved base areaHow to calculate areaVolume generated by a revolving Bernoulli Lemniscate - Integral boundary questionRelation between surface area and volume; and perimeter and area.gabriels horn: find a p for a p-series in such a way that the volume and surface area are infiniteVolume of a solid defined by two curves rotating around $x$-axis.
How to install OpenCV on Raspbian Stretch?
How to check if all elements of 1 list are in the *same quantity* and in any order, in the list2?
Can a Bladesinger Wizard use Bladesong with a Hand Crossbow?
What connection does MS Office have to Netscape Navigator?
Why do airplanes bank sharply to the right after air-to-air refueling?
Math-accent symbol over parentheses enclosing accented symbol (amsmath)
0 rank tensor vs 1D vector
Is it professional to write unrelated content in an almost-empty email?
Bartok - Syncopation (1): Meaning of notes in between Grand Staff
Is micro rebar a better way to reinforce concrete than rebar?
Why did CATV standarize in 75 ohms and everyone else in 50?
How to get from Geneva Airport to Metabief?
RigExpert AA-35 - Interpreting The Information
Why is information "lost" when it got into a black hole?
How to avoid supervisors with prejudiced views?
Are police here, aren't itthey?
What flight has the highest ratio of time difference to flight time?
Method for adding error messages to a dictionary given a key
What does "Its cash flow is deeply negative" mean?
Would a grinding machine be a simple and workable propulsion system for an interplanetary spacecraft?
Why don't programming languages automatically manage the synchronous/asynchronous problem?
Example of a Mathematician/Physicist whose Other Publications during their PhD eclipsed their PhD Thesis
INSERT to a table from a database to other (same SQL Server) using Dynamic SQL
Why the difference in type-inference over the as-pattern in two similar function definitions?
Bernoulli Lemniscate - surface area and volume
The Next CEO of Stack OverflowFind the volume of the solid obtained by rotating the region enclosed by the lines…Area of solid revolution using integration.Question on area under and between curves and volume of a solid by revolutionFind Volume of the Solid Obtained by Rotating Region(Disk Method)Cylinder volume with curved base areaHow to calculate areaVolume generated by a revolving Bernoulli Lemniscate - Integral boundary questionRelation between surface area and volume; and perimeter and area.gabriels horn: find a p for a p-series in such a way that the volume and surface area are infiniteVolume of a solid defined by two curves rotating around $x$-axis.
$begingroup$
How do I calculate the surface area and volume of the solid obtained by rotating the Bernoulli lemniscate
$$(x^2+y^2)^2=2a^2(x^2-y^2)$$
around the $x$-axis?
It is not like I'm lazy and asking for a ready solution, or completely helpless. I really tried to calculate this and I failed. That is why I'm asking for help :)
calculus volume area
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
$endgroup$
add a comment |
$begingroup$
How do I calculate the surface area and volume of the solid obtained by rotating the Bernoulli lemniscate
$$(x^2+y^2)^2=2a^2(x^2-y^2)$$
around the $x$-axis?
It is not like I'm lazy and asking for a ready solution, or completely helpless. I really tried to calculate this and I failed. That is why I'm asking for help :)
calculus volume area
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
$endgroup$
1
$begingroup$
Hi and welcome to the site. Could help if you added some details. Not everyone knows what a Bernoulli lemniscate is.
$endgroup$
– mathreadler
Sep 4 '16 at 18:02
1
$begingroup$
What counts as a solution, e.g,., is the definite integral formula for surface area and volume not suitable for you? After all, it's not as if the arc length of the lemniscate is some familiar number (a new letter can be introduced for it, but that's not quite solving it).
$endgroup$
– KCd
Sep 4 '16 at 18:03
add a comment |
$begingroup$
How do I calculate the surface area and volume of the solid obtained by rotating the Bernoulli lemniscate
$$(x^2+y^2)^2=2a^2(x^2-y^2)$$
around the $x$-axis?
It is not like I'm lazy and asking for a ready solution, or completely helpless. I really tried to calculate this and I failed. That is why I'm asking for help :)
calculus volume area
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
$endgroup$
How do I calculate the surface area and volume of the solid obtained by rotating the Bernoulli lemniscate
$$(x^2+y^2)^2=2a^2(x^2-y^2)$$
around the $x$-axis?
It is not like I'm lazy and asking for a ready solution, or completely helpless. I really tried to calculate this and I failed. That is why I'm asking for help :)
calculus volume area
calculus volume area
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
edited Sep 4 '16 at 19:19
Parcly Taxel
44.7k1376109
44.7k1376109
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
asked Sep 4 '16 at 17:59
Lucky_PierreLucky_Pierre
11
11
1
$begingroup$
Hi and welcome to the site. Could help if you added some details. Not everyone knows what a Bernoulli lemniscate is.
$endgroup$
– mathreadler
Sep 4 '16 at 18:02
1
$begingroup$
What counts as a solution, e.g,., is the definite integral formula for surface area and volume not suitable for you? After all, it's not as if the arc length of the lemniscate is some familiar number (a new letter can be introduced for it, but that's not quite solving it).
$endgroup$
– KCd
Sep 4 '16 at 18:03
add a comment |
1
$begingroup$
Hi and welcome to the site. Could help if you added some details. Not everyone knows what a Bernoulli lemniscate is.
$endgroup$
– mathreadler
Sep 4 '16 at 18:02
1
$begingroup$
What counts as a solution, e.g,., is the definite integral formula for surface area and volume not suitable for you? After all, it's not as if the arc length of the lemniscate is some familiar number (a new letter can be introduced for it, but that's not quite solving it).
$endgroup$
– KCd
Sep 4 '16 at 18:03
1
1
$begingroup$
Hi and welcome to the site. Could help if you added some details. Not everyone knows what a Bernoulli lemniscate is.
$endgroup$
– mathreadler
Sep 4 '16 at 18:02
$begingroup$
Hi and welcome to the site. Could help if you added some details. Not everyone knows what a Bernoulli lemniscate is.
$endgroup$
– mathreadler
Sep 4 '16 at 18:02
1
1
$begingroup$
What counts as a solution, e.g,., is the definite integral formula for surface area and volume not suitable for you? After all, it's not as if the arc length of the lemniscate is some familiar number (a new letter can be introduced for it, but that's not quite solving it).
$endgroup$
– KCd
Sep 4 '16 at 18:03
$begingroup$
What counts as a solution, e.g,., is the definite integral formula for surface area and volume not suitable for you? After all, it's not as if the arc length of the lemniscate is some familiar number (a new letter can be introduced for it, but that's not quite solving it).
$endgroup$
– KCd
Sep 4 '16 at 18:03
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The solution in the polar coordinates system where the lemniscate is given by the formula $r^2=2a^2cos2phi$.
Surface area can be obtained by using the formula $A=2piint_a^br(phi)sinphisqrtr^2(phi)+[dr(phi)/dphi]^2dphi$:
$$A=2pi a^2sqrtaint_a^bsinphisqrtcos2phi +sin^22phi dphi$$
The integral above seems to have no analytical solutions.
The formula for the volume is pretty simple and can be obtained directly from the formula $V=piint_a^b r^2(phi)dphi$ :
$$V=2a^2piint_a^b cos2phi dphi$$
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
$endgroup$
add a comment |
$begingroup$
See the image (https://i.stack.imgur.com/a6cBL.png).
Equation -: (x^2 + y^2)^2 = r^2*(x^2 - y^2).
If the maximum distance from the center of the Lemniscate of Bernoulli (origin) to the end point on x-axis is 'r', then Volume obtained will be 0.45536*(r^3) and Surface Area will be 1.36608*(r^2).
For proof, mail me at rasikrastogi@gmail.com .
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
$endgroup$
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%2f1914462%2fbernoulli-lemniscate-surface-area-and-volume%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
$begingroup$
The solution in the polar coordinates system where the lemniscate is given by the formula $r^2=2a^2cos2phi$.
Surface area can be obtained by using the formula $A=2piint_a^br(phi)sinphisqrtr^2(phi)+[dr(phi)/dphi]^2dphi$:
$$A=2pi a^2sqrtaint_a^bsinphisqrtcos2phi +sin^22phi dphi$$
The integral above seems to have no analytical solutions.
The formula for the volume is pretty simple and can be obtained directly from the formula $V=piint_a^b r^2(phi)dphi$ :
$$V=2a^2piint_a^b cos2phi dphi$$
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
$endgroup$
add a comment |
$begingroup$
The solution in the polar coordinates system where the lemniscate is given by the formula $r^2=2a^2cos2phi$.
Surface area can be obtained by using the formula $A=2piint_a^br(phi)sinphisqrtr^2(phi)+[dr(phi)/dphi]^2dphi$:
$$A=2pi a^2sqrtaint_a^bsinphisqrtcos2phi +sin^22phi dphi$$
The integral above seems to have no analytical solutions.
The formula for the volume is pretty simple and can be obtained directly from the formula $V=piint_a^b r^2(phi)dphi$ :
$$V=2a^2piint_a^b cos2phi dphi$$
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
$endgroup$
add a comment |
$begingroup$
The solution in the polar coordinates system where the lemniscate is given by the formula $r^2=2a^2cos2phi$.
Surface area can be obtained by using the formula $A=2piint_a^br(phi)sinphisqrtr^2(phi)+[dr(phi)/dphi]^2dphi$:
$$A=2pi a^2sqrtaint_a^bsinphisqrtcos2phi +sin^22phi dphi$$
The integral above seems to have no analytical solutions.
The formula for the volume is pretty simple and can be obtained directly from the formula $V=piint_a^b r^2(phi)dphi$ :
$$V=2a^2piint_a^b cos2phi dphi$$
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
$endgroup$
The solution in the polar coordinates system where the lemniscate is given by the formula $r^2=2a^2cos2phi$.
Surface area can be obtained by using the formula $A=2piint_a^br(phi)sinphisqrtr^2(phi)+[dr(phi)/dphi]^2dphi$:
$$A=2pi a^2sqrtaint_a^bsinphisqrtcos2phi +sin^22phi dphi$$
The integral above seems to have no analytical solutions.
The formula for the volume is pretty simple and can be obtained directly from the formula $V=piint_a^b r^2(phi)dphi$ :
$$V=2a^2piint_a^b cos2phi dphi$$
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
edited Sep 5 '16 at 19:00
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
answered Sep 5 '16 at 18:44
Lucky_PierreLucky_Pierre
11
11
add a comment |
add a comment |
$begingroup$
See the image (https://i.stack.imgur.com/a6cBL.png).
Equation -: (x^2 + y^2)^2 = r^2*(x^2 - y^2).
If the maximum distance from the center of the Lemniscate of Bernoulli (origin) to the end point on x-axis is 'r', then Volume obtained will be 0.45536*(r^3) and Surface Area will be 1.36608*(r^2).
For proof, mail me at rasikrastogi@gmail.com .
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
$endgroup$
add a comment |
$begingroup$
See the image (https://i.stack.imgur.com/a6cBL.png).
Equation -: (x^2 + y^2)^2 = r^2*(x^2 - y^2).
If the maximum distance from the center of the Lemniscate of Bernoulli (origin) to the end point on x-axis is 'r', then Volume obtained will be 0.45536*(r^3) and Surface Area will be 1.36608*(r^2).
For proof, mail me at rasikrastogi@gmail.com .
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
$endgroup$
add a comment |
$begingroup$
See the image (https://i.stack.imgur.com/a6cBL.png).
Equation -: (x^2 + y^2)^2 = r^2*(x^2 - y^2).
If the maximum distance from the center of the Lemniscate of Bernoulli (origin) to the end point on x-axis is 'r', then Volume obtained will be 0.45536*(r^3) and Surface Area will be 1.36608*(r^2).
For proof, mail me at rasikrastogi@gmail.com .
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
$endgroup$
See the image (https://i.stack.imgur.com/a6cBL.png).
Equation -: (x^2 + y^2)^2 = r^2*(x^2 - y^2).
If the maximum distance from the center of the Lemniscate of Bernoulli (origin) to the end point on x-axis is 'r', then Volume obtained will be 0.45536*(r^3) and Surface Area will be 1.36608*(r^2).
For proof, mail me at rasikrastogi@gmail.com .
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
answered Apr 17 '17 at 9:03
Rasik RastogiRasik Rastogi
1
1
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%2f1914462%2fbernoulli-lemniscate-surface-area-and-volume%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
1
$begingroup$
Hi and welcome to the site. Could help if you added some details. Not everyone knows what a Bernoulli lemniscate is.
$endgroup$
– mathreadler
Sep 4 '16 at 18:02
1
$begingroup$
What counts as a solution, e.g,., is the definite integral formula for surface area and volume not suitable for you? After all, it's not as if the arc length of the lemniscate is some familiar number (a new letter can be introduced for it, but that's not quite solving it).
$endgroup$
– KCd
Sep 4 '16 at 18:03