Surface integral and parametrization











up vote
0
down vote

favorite












I'm struggling with surface integrals, and I still do not have much confidence with the parameterization of functions. This is the exercise I would like to solve:




Calculate the surface integral of the function



$$f = (x-1)^2 + (y-2)^2$$



extended to the surface $P equiv (1+rho cos vartheta, 2 + rho sin vartheta, 4 - rho^2)$ with $vartheta in [0, 2pi], rho in [sqrt{2}, sqrt{3}]$




without going into detail in the steps and in the final result, how should I proceed? The first thing I should do is find the parameterization of the surface, right? And can I retrieve it?



Thanks in advances






surface-integrals







share|cite|improve this question


























    up vote
    0
    down vote

    favorite












    I'm struggling with surface integrals, and I still do not have much confidence with the parameterization of functions. This is the exercise I would like to solve:




    Calculate the surface integral of the function



    $$f = (x-1)^2 + (y-2)^2$$



    extended to the surface $P equiv (1+rho cos vartheta, 2 + rho sin vartheta, 4 - rho^2)$ with $vartheta in [0, 2pi], rho in [sqrt{2}, sqrt{3}]$




    without going into detail in the steps and in the final result, how should I proceed? The first thing I should do is find the parameterization of the surface, right? And can I retrieve it?



    Thanks in advances






    surface-integrals







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I'm struggling with surface integrals, and I still do not have much confidence with the parameterization of functions. This is the exercise I would like to solve:




      Calculate the surface integral of the function



      $$f = (x-1)^2 + (y-2)^2$$



      extended to the surface $P equiv (1+rho cos vartheta, 2 + rho sin vartheta, 4 - rho^2)$ with $vartheta in [0, 2pi], rho in [sqrt{2}, sqrt{3}]$




      without going into detail in the steps and in the final result, how should I proceed? The first thing I should do is find the parameterization of the surface, right? And can I retrieve it?



      Thanks in advances






      surface-integrals







      share|cite|improve this question













      I'm struggling with surface integrals, and I still do not have much confidence with the parameterization of functions. This is the exercise I would like to solve:




      Calculate the surface integral of the function



      $$f = (x-1)^2 + (y-2)^2$$



      extended to the surface $P equiv (1+rho cos vartheta, 2 + rho sin vartheta, 4 - rho^2)$ with $vartheta in [0, 2pi], rho in [sqrt{2}, sqrt{3}]$




      without going into detail in the steps and in the final result, how should I proceed? The first thing I should do is find the parameterization of the surface, right? And can I retrieve it?



      Thanks in advances






      surface-integrals




      surface-integrals






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 days ago









      user3204810

      1806




      1806






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          Your surface $S$ is already given in a parametrized form, i.e. your parametrization is
          $$ P(u,v) = (1+ucos v, 2+usin v, 4-v^2), quad uin [sqrt 2, sqrt 3], quad v in [0,2pi]. $$
          Now you should compute your surface area element $mathrm dA$, which is defined as
          $$ mathrm dA = |partial_u P times partial_v P| , mathrm du , mathrm dv, $$
          and the sought integral then becomes
          $$ int_S f , mathrm dA = int_{0}^{2pi} int_{sqrt 2}^{sqrt 3} f(P(u,v)) |partial_u P times partial_v P| , mathrm du ,mathrm dv. $$
          Can you proceed?






          share|cite|improve this answer

















          • 1




            Tomorrow I try calmly. In any case, thanks for the tip!
            – user3204810
            2 days ago











          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',
          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
          });


          }
          });














           

          draft saved


          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005405%2fsurface-integral-and-parametrization%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








          up vote
          1
          down vote



          accepted










          Your surface $S$ is already given in a parametrized form, i.e. your parametrization is
          $$ P(u,v) = (1+ucos v, 2+usin v, 4-v^2), quad uin [sqrt 2, sqrt 3], quad v in [0,2pi]. $$
          Now you should compute your surface area element $mathrm dA$, which is defined as
          $$ mathrm dA = |partial_u P times partial_v P| , mathrm du , mathrm dv, $$
          and the sought integral then becomes
          $$ int_S f , mathrm dA = int_{0}^{2pi} int_{sqrt 2}^{sqrt 3} f(P(u,v)) |partial_u P times partial_v P| , mathrm du ,mathrm dv. $$
          Can you proceed?






          share|cite|improve this answer

















          • 1




            Tomorrow I try calmly. In any case, thanks for the tip!
            – user3204810
            2 days ago















          up vote
          1
          down vote



          accepted










          Your surface $S$ is already given in a parametrized form, i.e. your parametrization is
          $$ P(u,v) = (1+ucos v, 2+usin v, 4-v^2), quad uin [sqrt 2, sqrt 3], quad v in [0,2pi]. $$
          Now you should compute your surface area element $mathrm dA$, which is defined as
          $$ mathrm dA = |partial_u P times partial_v P| , mathrm du , mathrm dv, $$
          and the sought integral then becomes
          $$ int_S f , mathrm dA = int_{0}^{2pi} int_{sqrt 2}^{sqrt 3} f(P(u,v)) |partial_u P times partial_v P| , mathrm du ,mathrm dv. $$
          Can you proceed?






          share|cite|improve this answer

















          • 1




            Tomorrow I try calmly. In any case, thanks for the tip!
            – user3204810
            2 days ago













          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          Your surface $S$ is already given in a parametrized form, i.e. your parametrization is
          $$ P(u,v) = (1+ucos v, 2+usin v, 4-v^2), quad uin [sqrt 2, sqrt 3], quad v in [0,2pi]. $$
          Now you should compute your surface area element $mathrm dA$, which is defined as
          $$ mathrm dA = |partial_u P times partial_v P| , mathrm du , mathrm dv, $$
          and the sought integral then becomes
          $$ int_S f , mathrm dA = int_{0}^{2pi} int_{sqrt 2}^{sqrt 3} f(P(u,v)) |partial_u P times partial_v P| , mathrm du ,mathrm dv. $$
          Can you proceed?






          share|cite|improve this answer












          Your surface $S$ is already given in a parametrized form, i.e. your parametrization is
          $$ P(u,v) = (1+ucos v, 2+usin v, 4-v^2), quad uin [sqrt 2, sqrt 3], quad v in [0,2pi]. $$
          Now you should compute your surface area element $mathrm dA$, which is defined as
          $$ mathrm dA = |partial_u P times partial_v P| , mathrm du , mathrm dv, $$
          and the sought integral then becomes
          $$ int_S f , mathrm dA = int_{0}^{2pi} int_{sqrt 2}^{sqrt 3} f(P(u,v)) |partial_u P times partial_v P| , mathrm du ,mathrm dv. $$
          Can you proceed?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 days ago









          MisterRiemann

          4,4751622




          4,4751622








          • 1




            Tomorrow I try calmly. In any case, thanks for the tip!
            – user3204810
            2 days ago














          • 1




            Tomorrow I try calmly. In any case, thanks for the tip!
            – user3204810
            2 days ago








          1




          1




          Tomorrow I try calmly. In any case, thanks for the tip!
          – user3204810
          2 days ago




          Tomorrow I try calmly. In any case, thanks for the tip!
          – user3204810
          2 days ago


















           

          draft saved


          draft discarded



















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005405%2fsurface-integral-and-parametrization%23new-answer', 'question_page');
          }
          );

          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







          Popular posts from this blog

          android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

          Image
          .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 This is a simple project showing cubes on Unity, exported to android project and imported on Android studio. On maifest file it showed following warning: Expecting <uses-feature android:name="android.software.leanback" android:required="false" /> tag I have no information about it. And also on default emptyview android project does not use/mention this feature. I know it is required by TV's and such but why its not added by default? Why is this? And Should I concern about it? android unity3d
          Read more

          SQL update select statement

          Image
          0 I'm trying to update an entry in my database (postgresql). I'm having a problem with updating the query: INSERT INTO prices (price, product_id) SELECT product_price, commn_id FROM products_temp as prod WHERE NOT EXISTS (SELECT * FROM prices od WHERE prod.commn_id = od.product_id) OR prod.product_price != ( SELECT price FROM prices as p WHERE p.product_id = prod.commn_id order by p.end desc LIMIT 1 ) Query works fine when i delete: OR prod.product_price != ( SELECT price FROM prices as p WHERE p.product_id = prod.commn_id order by p.end desc LIMIT 1 ) So it seems to me that it's looping through this operation. My question is, how can I fix it? sql postgresql
          Read more

          'app-layout' is not a known element: how to share Component with different Modules

          Image
          0 My application was working fine until I decided to go with lazy loading: So, my shared component looks like this: import { Component, Renderer2 } from '@angular/core'; export interface FormModel { captcha?: string; } @Component({ selector: 'app-layout', templateUrl: './app-layout.component.html', styleUrls: ['./app-layout.css'] }) export class FullLayoutComponent { } app-layout.component.ts I am using this component in my Another component say school-home.component.html like this: <app-layout> </app-layout> and it was working fine until I created a Module school-home.module.ts like this: import { NgModule } from '@angular/core'; import { CommonModule } from '@angular/common'; import { SchoolhomeRoutingModule
          Read more