Finding the circumference of a double cycloid












1












$begingroup$


So I have the following problem:



Find the the circumference of a melon, if the boundary line of the melon is a double cycloid and following equation describe the boundary line of the melon.



{x(t)=5 (t-sin(t)), y(t)=5(1-cos(t))} and



{x(t)=5 (t-sin(t)), y(t)=-5(1-cos(t))}



I have to give the following:



1) integrand for the one side of the melon: $$ g(t)= $$
2)indefinite integral for the half of the melon's circumference: $$ C(t)= ... + C $$
3) the circumference of the melon: $$ C=$$



I don't know how to find the upper and lower bound. So it would be very helpful, if somebody could give me the equations to draw a graph in geogebra or in wolframalpha.



What I have found out:
1) integrand for the one side of the melon: $$ g(t)=sqrt{50-50cos(t)} $$



3) the circumference of the melon: it should be $$ C=80$$, but if I take t from $ 0$ to $ 2pi $ $ g(t)=sqrt{50-50cos(t)} $ the answer comes 0. What is wrong?






calculus integration definite-integrals indefinite-integrals







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    So I have the following problem:



    Find the the circumference of a melon, if the boundary line of the melon is a double cycloid and following equation describe the boundary line of the melon.



    {x(t)=5 (t-sin(t)), y(t)=5(1-cos(t))} and



    {x(t)=5 (t-sin(t)), y(t)=-5(1-cos(t))}



    I have to give the following:



    1) integrand for the one side of the melon: $$ g(t)= $$
    2)indefinite integral for the half of the melon's circumference: $$ C(t)= ... + C $$
    3) the circumference of the melon: $$ C=$$



    I don't know how to find the upper and lower bound. So it would be very helpful, if somebody could give me the equations to draw a graph in geogebra or in wolframalpha.



    What I have found out:
    1) integrand for the one side of the melon: $$ g(t)=sqrt{50-50cos(t)} $$



    3) the circumference of the melon: it should be $$ C=80$$, but if I take t from $ 0$ to $ 2pi $ $ g(t)=sqrt{50-50cos(t)} $ the answer comes 0. What is wrong?






    calculus integration definite-integrals indefinite-integrals







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      So I have the following problem:



      Find the the circumference of a melon, if the boundary line of the melon is a double cycloid and following equation describe the boundary line of the melon.



      {x(t)=5 (t-sin(t)), y(t)=5(1-cos(t))} and



      {x(t)=5 (t-sin(t)), y(t)=-5(1-cos(t))}



      I have to give the following:



      1) integrand for the one side of the melon: $$ g(t)= $$
      2)indefinite integral for the half of the melon's circumference: $$ C(t)= ... + C $$
      3) the circumference of the melon: $$ C=$$



      I don't know how to find the upper and lower bound. So it would be very helpful, if somebody could give me the equations to draw a graph in geogebra or in wolframalpha.



      What I have found out:
      1) integrand for the one side of the melon: $$ g(t)=sqrt{50-50cos(t)} $$



      3) the circumference of the melon: it should be $$ C=80$$, but if I take t from $ 0$ to $ 2pi $ $ g(t)=sqrt{50-50cos(t)} $ the answer comes 0. What is wrong?






      calculus integration definite-integrals indefinite-integrals







      share|cite|improve this question











      $endgroup$




      So I have the following problem:



      Find the the circumference of a melon, if the boundary line of the melon is a double cycloid and following equation describe the boundary line of the melon.



      {x(t)=5 (t-sin(t)), y(t)=5(1-cos(t))} and



      {x(t)=5 (t-sin(t)), y(t)=-5(1-cos(t))}



      I have to give the following:



      1) integrand for the one side of the melon: $$ g(t)= $$
      2)indefinite integral for the half of the melon's circumference: $$ C(t)= ... + C $$
      3) the circumference of the melon: $$ C=$$



      I don't know how to find the upper and lower bound. So it would be very helpful, if somebody could give me the equations to draw a graph in geogebra or in wolframalpha.



      What I have found out:
      1) integrand for the one side of the melon: $$ g(t)=sqrt{50-50cos(t)} $$



      3) the circumference of the melon: it should be $$ C=80$$, but if I take t from $ 0$ to $ 2pi $ $ g(t)=sqrt{50-50cos(t)} $ the answer comes 0. What is wrong?






      calculus integration definite-integrals indefinite-integrals




      calculus integration definite-integrals indefinite-integrals






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 6 '18 at 10:18







      Student123

















      asked Dec 3 '18 at 14:16









      Student123Student123

      536




      536






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          This might help, so I am going to put it here.
          Calculate the length of a cycloid curve with a radius of 5 inches. The cycloid parametric equations are $x(t)=r(t-sin(t))$ and $y(t)=r(1-cos(t))$. $$f(t)=sqrt{left(frac{dx}{dt}right)^{2}+left(frac{dy}{dt}right)^{2}}dt=sqrt{r^{2}(t-sin(t))^{2}+r^{2}(1-cos(t))^{2}}dt$$ $$=rcdotsqrt{2-2cos(t)}dt$$ Normally, I would stop at this point and integrate, but this can be beneficially simplified using a trig identity. $cos(theta)=1-2sin^{2}(theta/2)$ Factoring the radical gets $sqrt{2(1-cos(t)).}$ Applying the identity gets $sqrt{2(1-(1-2sin^{2}(t/2))}=sqrt{4sin^{2}(t/2)}=2sin(frac{t}{2})$. $$f(t)=rcdot2sin(frac{t}{2})dt$$ This last eqn is a great deal easier to integrate. $$int_{0}^{2pi}rcdot2sin(frac{t}{2})dt=-rcdot4cos(frac{t}{2})Big|_{0}^{2pi}=8r$$






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            As far as i can tell, no. Half of the circumference is what I show in the answer.
            $endgroup$
            – Narlin
            Dec 3 '18 at 15:14










          • $begingroup$
            where do you get the $cos(theta)=1-2sin^{2}(theta/2)$ from?
            $endgroup$
            – Student123
            Dec 6 '18 at 10:50










          • $begingroup$
            sosmath.com/trig/douangl/douangl.html about half way down the page.
            $endgroup$
            – Narlin
            Dec 6 '18 at 20:55











          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
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3024106%2ffinding-the-circumference-of-a-double-cycloid%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









          1












          $begingroup$

          This might help, so I am going to put it here.
          Calculate the length of a cycloid curve with a radius of 5 inches. The cycloid parametric equations are $x(t)=r(t-sin(t))$ and $y(t)=r(1-cos(t))$. $$f(t)=sqrt{left(frac{dx}{dt}right)^{2}+left(frac{dy}{dt}right)^{2}}dt=sqrt{r^{2}(t-sin(t))^{2}+r^{2}(1-cos(t))^{2}}dt$$ $$=rcdotsqrt{2-2cos(t)}dt$$ Normally, I would stop at this point and integrate, but this can be beneficially simplified using a trig identity. $cos(theta)=1-2sin^{2}(theta/2)$ Factoring the radical gets $sqrt{2(1-cos(t)).}$ Applying the identity gets $sqrt{2(1-(1-2sin^{2}(t/2))}=sqrt{4sin^{2}(t/2)}=2sin(frac{t}{2})$. $$f(t)=rcdot2sin(frac{t}{2})dt$$ This last eqn is a great deal easier to integrate. $$int_{0}^{2pi}rcdot2sin(frac{t}{2})dt=-rcdot4cos(frac{t}{2})Big|_{0}^{2pi}=8r$$






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            As far as i can tell, no. Half of the circumference is what I show in the answer.
            $endgroup$
            – Narlin
            Dec 3 '18 at 15:14










          • $begingroup$
            where do you get the $cos(theta)=1-2sin^{2}(theta/2)$ from?
            $endgroup$
            – Student123
            Dec 6 '18 at 10:50










          • $begingroup$
            sosmath.com/trig/douangl/douangl.html about half way down the page.
            $endgroup$
            – Narlin
            Dec 6 '18 at 20:55
















          1












          $begingroup$

          This might help, so I am going to put it here.
          Calculate the length of a cycloid curve with a radius of 5 inches. The cycloid parametric equations are $x(t)=r(t-sin(t))$ and $y(t)=r(1-cos(t))$. $$f(t)=sqrt{left(frac{dx}{dt}right)^{2}+left(frac{dy}{dt}right)^{2}}dt=sqrt{r^{2}(t-sin(t))^{2}+r^{2}(1-cos(t))^{2}}dt$$ $$=rcdotsqrt{2-2cos(t)}dt$$ Normally, I would stop at this point and integrate, but this can be beneficially simplified using a trig identity. $cos(theta)=1-2sin^{2}(theta/2)$ Factoring the radical gets $sqrt{2(1-cos(t)).}$ Applying the identity gets $sqrt{2(1-(1-2sin^{2}(t/2))}=sqrt{4sin^{2}(t/2)}=2sin(frac{t}{2})$. $$f(t)=rcdot2sin(frac{t}{2})dt$$ This last eqn is a great deal easier to integrate. $$int_{0}^{2pi}rcdot2sin(frac{t}{2})dt=-rcdot4cos(frac{t}{2})Big|_{0}^{2pi}=8r$$






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            As far as i can tell, no. Half of the circumference is what I show in the answer.
            $endgroup$
            – Narlin
            Dec 3 '18 at 15:14










          • $begingroup$
            where do you get the $cos(theta)=1-2sin^{2}(theta/2)$ from?
            $endgroup$
            – Student123
            Dec 6 '18 at 10:50










          • $begingroup$
            sosmath.com/trig/douangl/douangl.html about half way down the page.
            $endgroup$
            – Narlin
            Dec 6 '18 at 20:55














          1












          1








          1





          $begingroup$

          This might help, so I am going to put it here.
          Calculate the length of a cycloid curve with a radius of 5 inches. The cycloid parametric equations are $x(t)=r(t-sin(t))$ and $y(t)=r(1-cos(t))$. $$f(t)=sqrt{left(frac{dx}{dt}right)^{2}+left(frac{dy}{dt}right)^{2}}dt=sqrt{r^{2}(t-sin(t))^{2}+r^{2}(1-cos(t))^{2}}dt$$ $$=rcdotsqrt{2-2cos(t)}dt$$ Normally, I would stop at this point and integrate, but this can be beneficially simplified using a trig identity. $cos(theta)=1-2sin^{2}(theta/2)$ Factoring the radical gets $sqrt{2(1-cos(t)).}$ Applying the identity gets $sqrt{2(1-(1-2sin^{2}(t/2))}=sqrt{4sin^{2}(t/2)}=2sin(frac{t}{2})$. $$f(t)=rcdot2sin(frac{t}{2})dt$$ This last eqn is a great deal easier to integrate. $$int_{0}^{2pi}rcdot2sin(frac{t}{2})dt=-rcdot4cos(frac{t}{2})Big|_{0}^{2pi}=8r$$






          share|cite|improve this answer









          $endgroup$



          This might help, so I am going to put it here.
          Calculate the length of a cycloid curve with a radius of 5 inches. The cycloid parametric equations are $x(t)=r(t-sin(t))$ and $y(t)=r(1-cos(t))$. $$f(t)=sqrt{left(frac{dx}{dt}right)^{2}+left(frac{dy}{dt}right)^{2}}dt=sqrt{r^{2}(t-sin(t))^{2}+r^{2}(1-cos(t))^{2}}dt$$ $$=rcdotsqrt{2-2cos(t)}dt$$ Normally, I would stop at this point and integrate, but this can be beneficially simplified using a trig identity. $cos(theta)=1-2sin^{2}(theta/2)$ Factoring the radical gets $sqrt{2(1-cos(t)).}$ Applying the identity gets $sqrt{2(1-(1-2sin^{2}(t/2))}=sqrt{4sin^{2}(t/2)}=2sin(frac{t}{2})$. $$f(t)=rcdot2sin(frac{t}{2})dt$$ This last eqn is a great deal easier to integrate. $$int_{0}^{2pi}rcdot2sin(frac{t}{2})dt=-rcdot4cos(frac{t}{2})Big|_{0}^{2pi}=8r$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 3 '18 at 14:51









          NarlinNarlin

          524310




          524310








          • 1




            $begingroup$
            As far as i can tell, no. Half of the circumference is what I show in the answer.
            $endgroup$
            – Narlin
            Dec 3 '18 at 15:14










          • $begingroup$
            where do you get the $cos(theta)=1-2sin^{2}(theta/2)$ from?
            $endgroup$
            – Student123
            Dec 6 '18 at 10:50










          • $begingroup$
            sosmath.com/trig/douangl/douangl.html about half way down the page.
            $endgroup$
            – Narlin
            Dec 6 '18 at 20:55














          • 1




            $begingroup$
            As far as i can tell, no. Half of the circumference is what I show in the answer.
            $endgroup$
            – Narlin
            Dec 3 '18 at 15:14










          • $begingroup$
            where do you get the $cos(theta)=1-2sin^{2}(theta/2)$ from?
            $endgroup$
            – Student123
            Dec 6 '18 at 10:50










          • $begingroup$
            sosmath.com/trig/douangl/douangl.html about half way down the page.
            $endgroup$
            – Narlin
            Dec 6 '18 at 20:55








          1




          1




          $begingroup$
          As far as i can tell, no. Half of the circumference is what I show in the answer.
          $endgroup$
          – Narlin
          Dec 3 '18 at 15:14




          $begingroup$
          As far as i can tell, no. Half of the circumference is what I show in the answer.
          $endgroup$
          – Narlin
          Dec 3 '18 at 15:14












          $begingroup$
          where do you get the $cos(theta)=1-2sin^{2}(theta/2)$ from?
          $endgroup$
          – Student123
          Dec 6 '18 at 10:50




          $begingroup$
          where do you get the $cos(theta)=1-2sin^{2}(theta/2)$ from?
          $endgroup$
          – Student123
          Dec 6 '18 at 10:50












          $begingroup$
          sosmath.com/trig/douangl/douangl.html about half way down the page.
          $endgroup$
          – Narlin
          Dec 6 '18 at 20:55




          $begingroup$
          sosmath.com/trig/douangl/douangl.html about half way down the page.
          $endgroup$
          – Narlin
          Dec 6 '18 at 20:55


















          draft saved

          draft discarded




















































          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.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3024106%2ffinding-the-circumference-of-a-double-cycloid%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

          Biblatex bibliography style without URLs when DOI exists (in Overleaf with Zotero bibliography)

          Image
          2 2 I have a Zotero bibliography in Overleaf. I use biblatex . usepackage[backend=biber,citestyle=authoryear]{biblatex} addbibresource{zoteroALH.bib} Bibliography entries look like: @book{de_saint-gervais_uniformization_2016, title = {Uniformization of {Riemann} {Surfaces}}, isbn = {978-3-03719-145-3}, url = {https://www.ems-ph.org/books/book.php?proj_nr=198&srch=series%7Chem}, urldate = {2019-02-03}, author = {de Saint-Gervais, Paul Henri}, year = {2016}, doi = {10.4171/145} } In the bibliography down the paper the reference shows like this: There is the DOI, great, but also the URL which is unnecessary when the DOI is there. How can I remove the URL when the DOI is present? biblatex bibliographies overleaf zote
          Read more

          ComboBox Display Member on multiple fields

          Image
          2 I'm trying to set the .DisplayMember property of a ComboBox in C#, but I want to bind it to multiple columns in the .DataSouce . My SQL looks like this: SELECT PersNbr, PersFirstName, PersMiddleName, PersLastName FROM Pers WHERE PersNbr = :persNbr; I'm saving this query in a DataTable so each column selected has it's own column in the Datatable . I want to make the .DisplayMember a combination of PersFirstName + PersMiddleName + PersLastName so their full name appears like this: comboBox.DisplayMemeber = "PersFirstName" + "PersMiddleName" + "PersLastName" I know I can just to this on the query: SELECT PersNbr, (PersFirstName || PersMiddleName || PersLastName) PersName and then just do this: comboBox.DisplayMember = "PersName"; but I
          Read more

          Is it possible to collect Nectar points via Trainline?

          Image
          2 Since I am not eligible for any Railcard, the only available discount/bonus scheme is Nectar Card for me. GWR, SWR, LNER and Virgin Trains allow to collect Nectar points and I have already linked a card to my GWR account. A friend suggested me to use Trainline App, because it can search for bus tickets either. I had a look at the application, but could not see any option for Nectar. Is there a way to collect Nectar points when I buy tickets via Trainline? uk trains loyalty-programs trainline share | improve this question asked Dec 2 '18 at 15:50 ahmedus 3,224 5 19 49
          Read more