Edit: Solved, Calculus homework: rewriting limit as definite integral












1












$begingroup$


On my Calculus homework, there is a question that I am having trouble with.



The question is:



Rewrite
$lim_{n to infty}sum_{i=1}^{n}frac{1}{n}left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^3right]$ as an integral $int_{a}^{b}f(x)dx$



I know how to do this kind of problems, but this one is a little confusing.



I know that:
$$Delta x = frac{b-a}{n}\x_i = Delta x i + a\lim_{n to infty}sum_{i=1}^{n}f(x_i)Delta x=int_{a}^{b}f(x)dx$$
and in the question
$$begin{eqnarray*}frac{1}{n} &=& frac{b-a}{n} = Delta x\1&=&b-a\a&=&0\b&=&1end{eqnarray*}$$
So the answer must be something like:
$$int_{0}^{1}f(x)dx$$
What I don't understand is the $f(x_i)$ part.



I don't know how to simplify the expression inside the bracket.



It becomes something like:



$$left[x^{3}+x^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$$



I can't figure out what $f(x)$ is.



Any help or explanations are appreciated, thanks.




Edit:
The question in my homework might have a typo. Thank you for your help.






calculus integration limits definite-integrals summation







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    On my Calculus homework, there is a question that I am having trouble with.



    The question is:



    Rewrite
    $lim_{n to infty}sum_{i=1}^{n}frac{1}{n}left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^3right]$ as an integral $int_{a}^{b}f(x)dx$



    I know how to do this kind of problems, but this one is a little confusing.



    I know that:
    $$Delta x = frac{b-a}{n}\x_i = Delta x i + a\lim_{n to infty}sum_{i=1}^{n}f(x_i)Delta x=int_{a}^{b}f(x)dx$$
    and in the question
    $$begin{eqnarray*}frac{1}{n} &=& frac{b-a}{n} = Delta x\1&=&b-a\a&=&0\b&=&1end{eqnarray*}$$
    So the answer must be something like:
    $$int_{0}^{1}f(x)dx$$
    What I don't understand is the $f(x_i)$ part.



    I don't know how to simplify the expression inside the bracket.



    It becomes something like:



    $$left[x^{3}+x^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$$



    I can't figure out what $f(x)$ is.



    Any help or explanations are appreciated, thanks.




    Edit:
    The question in my homework might have a typo. Thank you for your help.






    calculus integration limits definite-integrals summation







    share|cite|improve this question











    $endgroup$















      1












      1








      1


      0



      $begingroup$


      On my Calculus homework, there is a question that I am having trouble with.



      The question is:



      Rewrite
      $lim_{n to infty}sum_{i=1}^{n}frac{1}{n}left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^3right]$ as an integral $int_{a}^{b}f(x)dx$



      I know how to do this kind of problems, but this one is a little confusing.



      I know that:
      $$Delta x = frac{b-a}{n}\x_i = Delta x i + a\lim_{n to infty}sum_{i=1}^{n}f(x_i)Delta x=int_{a}^{b}f(x)dx$$
      and in the question
      $$begin{eqnarray*}frac{1}{n} &=& frac{b-a}{n} = Delta x\1&=&b-a\a&=&0\b&=&1end{eqnarray*}$$
      So the answer must be something like:
      $$int_{0}^{1}f(x)dx$$
      What I don't understand is the $f(x_i)$ part.



      I don't know how to simplify the expression inside the bracket.



      It becomes something like:



      $$left[x^{3}+x^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$$



      I can't figure out what $f(x)$ is.



      Any help or explanations are appreciated, thanks.




      Edit:
      The question in my homework might have a typo. Thank you for your help.






      calculus integration limits definite-integrals summation







      share|cite|improve this question











      $endgroup$




      On my Calculus homework, there is a question that I am having trouble with.



      The question is:



      Rewrite
      $lim_{n to infty}sum_{i=1}^{n}frac{1}{n}left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^3right]$ as an integral $int_{a}^{b}f(x)dx$



      I know how to do this kind of problems, but this one is a little confusing.



      I know that:
      $$Delta x = frac{b-a}{n}\x_i = Delta x i + a\lim_{n to infty}sum_{i=1}^{n}f(x_i)Delta x=int_{a}^{b}f(x)dx$$
      and in the question
      $$begin{eqnarray*}frac{1}{n} &=& frac{b-a}{n} = Delta x\1&=&b-a\a&=&0\b&=&1end{eqnarray*}$$
      So the answer must be something like:
      $$int_{0}^{1}f(x)dx$$
      What I don't understand is the $f(x_i)$ part.



      I don't know how to simplify the expression inside the bracket.



      It becomes something like:



      $$left[x^{3}+x^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$$



      I can't figure out what $f(x)$ is.



      Any help or explanations are appreciated, thanks.




      Edit:
      The question in my homework might have a typo. Thank you for your help.






      calculus integration limits definite-integrals summation




      calculus integration limits definite-integrals summation






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 9 at 5:57







      Yuchao

















      asked Jan 9 at 4:39









      YuchaoYuchao

      83




      83






















          1 Answer
          1






          active

          oldest

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          1












          $begingroup$

          Recall that we are evaluate rectangular area $f(x_i)cdot Delta x$, here $x_i = frac{i}{n}$ and then we sum up the rectangular areas.



          $$f(x)=x^3$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Hi and thank you for your answer. But can I ask how did you get $f(x)=x^{3}$ from the summation. I don't understand at all how $left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$ simplify into x^3.
            $endgroup$
            – Yuchao
            Jan 9 at 5:36












          • $begingroup$
            Actually upon closer examination, are you sure the question is not $sum frac1n left( frac{i}nright)^3$. You mean there is another summation inside the summation?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:39










          • $begingroup$
            The question is not $sum frac{1}{n}left(frac{i}{n}right)^{3}$. I don't think there is another summation inside the summation. It looks like there is a series inside the summation since there is an ellipsis.
            $endgroup$
            – Yuchao
            Jan 9 at 5:44












          • $begingroup$
            I've checked the answers from my homework. The correct answer is $f(x)=x^{3}$, but I'm not sure how the answer was found.
            $endgroup$
            – Yuchao
            Jan 9 at 5:45










          • $begingroup$
            I think there is a typo, the question intended to ask identity $lim_{n to infty}sum frac1n left( frac{i}{n}right)^3$ as an integral.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:47












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          1 Answer
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          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          Recall that we are evaluate rectangular area $f(x_i)cdot Delta x$, here $x_i = frac{i}{n}$ and then we sum up the rectangular areas.



          $$f(x)=x^3$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Hi and thank you for your answer. But can I ask how did you get $f(x)=x^{3}$ from the summation. I don't understand at all how $left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$ simplify into x^3.
            $endgroup$
            – Yuchao
            Jan 9 at 5:36












          • $begingroup$
            Actually upon closer examination, are you sure the question is not $sum frac1n left( frac{i}nright)^3$. You mean there is another summation inside the summation?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:39










          • $begingroup$
            The question is not $sum frac{1}{n}left(frac{i}{n}right)^{3}$. I don't think there is another summation inside the summation. It looks like there is a series inside the summation since there is an ellipsis.
            $endgroup$
            – Yuchao
            Jan 9 at 5:44












          • $begingroup$
            I've checked the answers from my homework. The correct answer is $f(x)=x^{3}$, but I'm not sure how the answer was found.
            $endgroup$
            – Yuchao
            Jan 9 at 5:45










          • $begingroup$
            I think there is a typo, the question intended to ask identity $lim_{n to infty}sum frac1n left( frac{i}{n}right)^3$ as an integral.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:47
















          1












          $begingroup$

          Recall that we are evaluate rectangular area $f(x_i)cdot Delta x$, here $x_i = frac{i}{n}$ and then we sum up the rectangular areas.



          $$f(x)=x^3$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Hi and thank you for your answer. But can I ask how did you get $f(x)=x^{3}$ from the summation. I don't understand at all how $left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$ simplify into x^3.
            $endgroup$
            – Yuchao
            Jan 9 at 5:36












          • $begingroup$
            Actually upon closer examination, are you sure the question is not $sum frac1n left( frac{i}nright)^3$. You mean there is another summation inside the summation?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:39










          • $begingroup$
            The question is not $sum frac{1}{n}left(frac{i}{n}right)^{3}$. I don't think there is another summation inside the summation. It looks like there is a series inside the summation since there is an ellipsis.
            $endgroup$
            – Yuchao
            Jan 9 at 5:44












          • $begingroup$
            I've checked the answers from my homework. The correct answer is $f(x)=x^{3}$, but I'm not sure how the answer was found.
            $endgroup$
            – Yuchao
            Jan 9 at 5:45










          • $begingroup$
            I think there is a typo, the question intended to ask identity $lim_{n to infty}sum frac1n left( frac{i}{n}right)^3$ as an integral.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:47














          1












          1








          1





          $begingroup$

          Recall that we are evaluate rectangular area $f(x_i)cdot Delta x$, here $x_i = frac{i}{n}$ and then we sum up the rectangular areas.



          $$f(x)=x^3$$






          share|cite|improve this answer









          $endgroup$



          Recall that we are evaluate rectangular area $f(x_i)cdot Delta x$, here $x_i = frac{i}{n}$ and then we sum up the rectangular areas.



          $$f(x)=x^3$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 9 at 5:31









          Siong Thye GohSiong Thye Goh

          104k1468120




          104k1468120












          • $begingroup$
            Hi and thank you for your answer. But can I ask how did you get $f(x)=x^{3}$ from the summation. I don't understand at all how $left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$ simplify into x^3.
            $endgroup$
            – Yuchao
            Jan 9 at 5:36












          • $begingroup$
            Actually upon closer examination, are you sure the question is not $sum frac1n left( frac{i}nright)^3$. You mean there is another summation inside the summation?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:39










          • $begingroup$
            The question is not $sum frac{1}{n}left(frac{i}{n}right)^{3}$. I don't think there is another summation inside the summation. It looks like there is a series inside the summation since there is an ellipsis.
            $endgroup$
            – Yuchao
            Jan 9 at 5:44












          • $begingroup$
            I've checked the answers from my homework. The correct answer is $f(x)=x^{3}$, but I'm not sure how the answer was found.
            $endgroup$
            – Yuchao
            Jan 9 at 5:45










          • $begingroup$
            I think there is a typo, the question intended to ask identity $lim_{n to infty}sum frac1n left( frac{i}{n}right)^3$ as an integral.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:47


















          • $begingroup$
            Hi and thank you for your answer. But can I ask how did you get $f(x)=x^{3}$ from the summation. I don't understand at all how $left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$ simplify into x^3.
            $endgroup$
            – Yuchao
            Jan 9 at 5:36












          • $begingroup$
            Actually upon closer examination, are you sure the question is not $sum frac1n left( frac{i}nright)^3$. You mean there is another summation inside the summation?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:39










          • $begingroup$
            The question is not $sum frac{1}{n}left(frac{i}{n}right)^{3}$. I don't think there is another summation inside the summation. It looks like there is a series inside the summation since there is an ellipsis.
            $endgroup$
            – Yuchao
            Jan 9 at 5:44












          • $begingroup$
            I've checked the answers from my homework. The correct answer is $f(x)=x^{3}$, but I'm not sure how the answer was found.
            $endgroup$
            – Yuchao
            Jan 9 at 5:45










          • $begingroup$
            I think there is a typo, the question intended to ask identity $lim_{n to infty}sum frac1n left( frac{i}{n}right)^3$ as an integral.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 5:47
















          $begingroup$
          Hi and thank you for your answer. But can I ask how did you get $f(x)=x^{3}$ from the summation. I don't understand at all how $left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$ simplify into x^3.
          $endgroup$
          – Yuchao
          Jan 9 at 5:36






          $begingroup$
          Hi and thank you for your answer. But can I ask how did you get $f(x)=x^{3}$ from the summation. I don't understand at all how $left[left(frac{i}{n}right)^{3}+left(frac{i}{n}right)^{3}+ldots+left(frac{n-1}{n}right)^{3}right]$ simplify into x^3.
          $endgroup$
          – Yuchao
          Jan 9 at 5:36














          $begingroup$
          Actually upon closer examination, are you sure the question is not $sum frac1n left( frac{i}nright)^3$. You mean there is another summation inside the summation?
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 5:39




          $begingroup$
          Actually upon closer examination, are you sure the question is not $sum frac1n left( frac{i}nright)^3$. You mean there is another summation inside the summation?
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 5:39












          $begingroup$
          The question is not $sum frac{1}{n}left(frac{i}{n}right)^{3}$. I don't think there is another summation inside the summation. It looks like there is a series inside the summation since there is an ellipsis.
          $endgroup$
          – Yuchao
          Jan 9 at 5:44






          $begingroup$
          The question is not $sum frac{1}{n}left(frac{i}{n}right)^{3}$. I don't think there is another summation inside the summation. It looks like there is a series inside the summation since there is an ellipsis.
          $endgroup$
          – Yuchao
          Jan 9 at 5:44














          $begingroup$
          I've checked the answers from my homework. The correct answer is $f(x)=x^{3}$, but I'm not sure how the answer was found.
          $endgroup$
          – Yuchao
          Jan 9 at 5:45




          $begingroup$
          I've checked the answers from my homework. The correct answer is $f(x)=x^{3}$, but I'm not sure how the answer was found.
          $endgroup$
          – Yuchao
          Jan 9 at 5:45












          $begingroup$
          I think there is a typo, the question intended to ask identity $lim_{n to infty}sum frac1n left( frac{i}{n}right)^3$ as an integral.
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 5:47




          $begingroup$
          I think there is a typo, the question intended to ask identity $lim_{n to infty}sum frac1n left( frac{i}{n}right)^3$ as an integral.
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 5:47


















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