Why $sum(x)$ is greater than $int x , dx$












0












$begingroup$


I studied that difference between integration and summation is Summation is evaluated on discrete values while integration is evaluated on continuous values. If we consider a strictly increasing function like $f(x)=x$, $sum_{i=1}^2 x=3$ is greater than $int_{i=1}^2 x=1.5$ but according to above statement Integration value should be more since in integration every small value is adding up to the sum. But it is not happening. Am I going wrong with basic definitions of Integration and summations?






integration summation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You can interpret the sum as an area under the ‘curve’. Try to figure out what is this curve in your example.
    $endgroup$
    – lcv
    Dec 24 '18 at 10:06










  • $begingroup$
    Here, curve is y=x and area made by y=x with x-axis between the interval 1 to 2 is 1.5 which is equal to integral of x from 1 to 2. But my doubt is, does integral of x from 1 to 2 mean summation of x for 1,1.000001,1.000002,....2 is same?
    $endgroup$
    – Sai Satwik Kuppili
    Dec 24 '18 at 10:15










  • $begingroup$
    @SaiSatwikKuppili no the curve is a staricase
    $endgroup$
    – lcv
    Dec 24 '18 at 10:29
















0












$begingroup$


I studied that difference between integration and summation is Summation is evaluated on discrete values while integration is evaluated on continuous values. If we consider a strictly increasing function like $f(x)=x$, $sum_{i=1}^2 x=3$ is greater than $int_{i=1}^2 x=1.5$ but according to above statement Integration value should be more since in integration every small value is adding up to the sum. But it is not happening. Am I going wrong with basic definitions of Integration and summations?






integration summation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You can interpret the sum as an area under the ‘curve’. Try to figure out what is this curve in your example.
    $endgroup$
    – lcv
    Dec 24 '18 at 10:06










  • $begingroup$
    Here, curve is y=x and area made by y=x with x-axis between the interval 1 to 2 is 1.5 which is equal to integral of x from 1 to 2. But my doubt is, does integral of x from 1 to 2 mean summation of x for 1,1.000001,1.000002,....2 is same?
    $endgroup$
    – Sai Satwik Kuppili
    Dec 24 '18 at 10:15










  • $begingroup$
    @SaiSatwikKuppili no the curve is a staricase
    $endgroup$
    – lcv
    Dec 24 '18 at 10:29














0












0








0





$begingroup$


I studied that difference between integration and summation is Summation is evaluated on discrete values while integration is evaluated on continuous values. If we consider a strictly increasing function like $f(x)=x$, $sum_{i=1}^2 x=3$ is greater than $int_{i=1}^2 x=1.5$ but according to above statement Integration value should be more since in integration every small value is adding up to the sum. But it is not happening. Am I going wrong with basic definitions of Integration and summations?






integration summation







share|cite|improve this question











$endgroup$




I studied that difference between integration and summation is Summation is evaluated on discrete values while integration is evaluated on continuous values. If we consider a strictly increasing function like $f(x)=x$, $sum_{i=1}^2 x=3$ is greater than $int_{i=1}^2 x=1.5$ but according to above statement Integration value should be more since in integration every small value is adding up to the sum. But it is not happening. Am I going wrong with basic definitions of Integration and summations?






integration summation




integration summation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 24 '18 at 10:15









Bernard

125k743118




125k743118










asked Dec 24 '18 at 10:01









Sai Satwik KuppiliSai Satwik Kuppili

658




658








  • 1




    $begingroup$
    You can interpret the sum as an area under the ‘curve’. Try to figure out what is this curve in your example.
    $endgroup$
    – lcv
    Dec 24 '18 at 10:06










  • $begingroup$
    Here, curve is y=x and area made by y=x with x-axis between the interval 1 to 2 is 1.5 which is equal to integral of x from 1 to 2. But my doubt is, does integral of x from 1 to 2 mean summation of x for 1,1.000001,1.000002,....2 is same?
    $endgroup$
    – Sai Satwik Kuppili
    Dec 24 '18 at 10:15










  • $begingroup$
    @SaiSatwikKuppili no the curve is a staricase
    $endgroup$
    – lcv
    Dec 24 '18 at 10:29














  • 1




    $begingroup$
    You can interpret the sum as an area under the ‘curve’. Try to figure out what is this curve in your example.
    $endgroup$
    – lcv
    Dec 24 '18 at 10:06










  • $begingroup$
    Here, curve is y=x and area made by y=x with x-axis between the interval 1 to 2 is 1.5 which is equal to integral of x from 1 to 2. But my doubt is, does integral of x from 1 to 2 mean summation of x for 1,1.000001,1.000002,....2 is same?
    $endgroup$
    – Sai Satwik Kuppili
    Dec 24 '18 at 10:15










  • $begingroup$
    @SaiSatwikKuppili no the curve is a staricase
    $endgroup$
    – lcv
    Dec 24 '18 at 10:29








1




1




$begingroup$
You can interpret the sum as an area under the ‘curve’. Try to figure out what is this curve in your example.
$endgroup$
– lcv
Dec 24 '18 at 10:06




$begingroup$
You can interpret the sum as an area under the ‘curve’. Try to figure out what is this curve in your example.
$endgroup$
– lcv
Dec 24 '18 at 10:06












$begingroup$
Here, curve is y=x and area made by y=x with x-axis between the interval 1 to 2 is 1.5 which is equal to integral of x from 1 to 2. But my doubt is, does integral of x from 1 to 2 mean summation of x for 1,1.000001,1.000002,....2 is same?
$endgroup$
– Sai Satwik Kuppili
Dec 24 '18 at 10:15




$begingroup$
Here, curve is y=x and area made by y=x with x-axis between the interval 1 to 2 is 1.5 which is equal to integral of x from 1 to 2. But my doubt is, does integral of x from 1 to 2 mean summation of x for 1,1.000001,1.000002,....2 is same?
$endgroup$
– Sai Satwik Kuppili
Dec 24 '18 at 10:15












$begingroup$
@SaiSatwikKuppili no the curve is a staricase
$endgroup$
– lcv
Dec 24 '18 at 10:29




$begingroup$
@SaiSatwikKuppili no the curve is a staricase
$endgroup$
– lcv
Dec 24 '18 at 10:29










2 Answers
2






active

oldest

votes


















1












$begingroup$

Actually you are treating unfair and $sum_{x=1}^2x$ (a sum of $2$ terms) should be compared with $int_0^2xdx$ (an integral over $2$ intervals of length $1$).



But also then: $$sum_{x=1}^2x=3>2=int_0^2xdx$$
More generally:
$$sum_{x=1}^nx=int_0^nlceil xrceil;dx>int_0^nx;dx$$ because $lceil xrceil> x$ almost everywhere.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    You want an explanation of $sum_{k=1}^n k>int_0^n x dx=sum_{k=1}^nint_{k=1}^k xdx$. It suffices to show $k>int_{k=1}^k xdx$, i.e. the height-$k$ width-$1$ rectangle is larger than the trapezium under $x$ from $x=k-1$ to $x=k$. From a diagram, that's trivial. More generally, integrating an increasing function gives such a rectangle-exceeds-area result.






    share|cite|improve this answer









    $endgroup$














      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
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051108%2fwhy-sumx-is-greater-than-int-x-dx%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









      1












      $begingroup$

      Actually you are treating unfair and $sum_{x=1}^2x$ (a sum of $2$ terms) should be compared with $int_0^2xdx$ (an integral over $2$ intervals of length $1$).



      But also then: $$sum_{x=1}^2x=3>2=int_0^2xdx$$
      More generally:
      $$sum_{x=1}^nx=int_0^nlceil xrceil;dx>int_0^nx;dx$$ because $lceil xrceil> x$ almost everywhere.






      share|cite|improve this answer









      $endgroup$


















        1












        $begingroup$

        Actually you are treating unfair and $sum_{x=1}^2x$ (a sum of $2$ terms) should be compared with $int_0^2xdx$ (an integral over $2$ intervals of length $1$).



        But also then: $$sum_{x=1}^2x=3>2=int_0^2xdx$$
        More generally:
        $$sum_{x=1}^nx=int_0^nlceil xrceil;dx>int_0^nx;dx$$ because $lceil xrceil> x$ almost everywhere.






        share|cite|improve this answer









        $endgroup$
















          1












          1








          1





          $begingroup$

          Actually you are treating unfair and $sum_{x=1}^2x$ (a sum of $2$ terms) should be compared with $int_0^2xdx$ (an integral over $2$ intervals of length $1$).



          But also then: $$sum_{x=1}^2x=3>2=int_0^2xdx$$
          More generally:
          $$sum_{x=1}^nx=int_0^nlceil xrceil;dx>int_0^nx;dx$$ because $lceil xrceil> x$ almost everywhere.






          share|cite|improve this answer









          $endgroup$



          Actually you are treating unfair and $sum_{x=1}^2x$ (a sum of $2$ terms) should be compared with $int_0^2xdx$ (an integral over $2$ intervals of length $1$).



          But also then: $$sum_{x=1}^2x=3>2=int_0^2xdx$$
          More generally:
          $$sum_{x=1}^nx=int_0^nlceil xrceil;dx>int_0^nx;dx$$ because $lceil xrceil> x$ almost everywhere.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 24 '18 at 10:25









          drhabdrhab

          104k545136




          104k545136























              0












              $begingroup$

              You want an explanation of $sum_{k=1}^n k>int_0^n x dx=sum_{k=1}^nint_{k=1}^k xdx$. It suffices to show $k>int_{k=1}^k xdx$, i.e. the height-$k$ width-$1$ rectangle is larger than the trapezium under $x$ from $x=k-1$ to $x=k$. From a diagram, that's trivial. More generally, integrating an increasing function gives such a rectangle-exceeds-area result.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                You want an explanation of $sum_{k=1}^n k>int_0^n x dx=sum_{k=1}^nint_{k=1}^k xdx$. It suffices to show $k>int_{k=1}^k xdx$, i.e. the height-$k$ width-$1$ rectangle is larger than the trapezium under $x$ from $x=k-1$ to $x=k$. From a diagram, that's trivial. More generally, integrating an increasing function gives such a rectangle-exceeds-area result.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  You want an explanation of $sum_{k=1}^n k>int_0^n x dx=sum_{k=1}^nint_{k=1}^k xdx$. It suffices to show $k>int_{k=1}^k xdx$, i.e. the height-$k$ width-$1$ rectangle is larger than the trapezium under $x$ from $x=k-1$ to $x=k$. From a diagram, that's trivial. More generally, integrating an increasing function gives such a rectangle-exceeds-area result.






                  share|cite|improve this answer









                  $endgroup$



                  You want an explanation of $sum_{k=1}^n k>int_0^n x dx=sum_{k=1}^nint_{k=1}^k xdx$. It suffices to show $k>int_{k=1}^k xdx$, i.e. the height-$k$ width-$1$ rectangle is larger than the trapezium under $x$ from $x=k-1$ to $x=k$. From a diagram, that's trivial. More generally, integrating an increasing function gives such a rectangle-exceeds-area result.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 24 '18 at 10:20









                  J.G.J.G.

                  34.2k23252




                  34.2k23252






























                      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%2f3051108%2fwhy-sumx-is-greater-than-int-x-dx%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

                      Plaza Victoria

                      Image
                      Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l
                      Read more

                      In PowerPoint, is there a keyboard shortcut for bulleted / numbered list?

                      Image
                      .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 2 OneNote has two useful keyboard shortcuts for bulleted and numbered lists, namely Ctrl + . and Ctrl + - . Is there something similar for PowerPoint where the work with lists is also very common? Haven't found anything here: Use keyboard shortcuts in PowerPoint. keyboard-shortcuts microsoft-powerpoint share | improve this question edited Aug 14 '14 at 14:49 Linger 2,826 10 28 40
                      Read more

                      How to put 3 figures in Latex with 2 figures side by side and 1 below these side by side images but in...

                      Image
                      1 0 I want to put 3 images in LateX such as 2 figures are side by side horizotally and 3rd figure below these 2 side by side figures but in middle. I have following code: begin{figure}[H] centering begin{minipage}{.5textwidth} centering includegraphics[width=.50linewidth]{Text/Images/Genelec_8010_AP.jpg} captionof{figure}{Genelec 8010 AP} label{fig:Genelec 8010} end{minipage}% begin{minipage}{.5textwidth} centering includegraphics[width=.69linewidth]{Text/Images/Genelec_8020_CPM.jpg} captionof{figure}{Genelec 8020 CPM} label{fig:Genelec 8020} end{minipage}% end{figure} %%Here is my 3rd picture code begin{figure} centering includegraphics[width=5cm]{Text/Images/Genelec_8030_BPM.jpg} caption{Genelec 8030 BPM} label{fig:Genelec 8030} end{figure} I want to ask ho
                      Read more