Calculate $int_{a}^{b}ln x ~dx$ using the definition of integral











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I am being asked to calculate the integral $int_{a}^{b}ln x~dx$ using the definition of integral (i.e. expressing it as limit of Riemann sums)



Here's what I did:
Let's divide the interval $[a,b]$ into $n$ subintervals of the form $[x_{i},x_{i+1}]$.



Let $x_i=a+frac{(b-a)i}{n}$ where $i=0,1,2...,n-1. $ be the start points of each subinterval such that $x_0=a$ and $x_n=b$



Let $Delta x_i=x_i-x_{i-1}=frac{b-a}{n}$



I defined the integral like this:



$$begin{align}int_{a}^{b}ln x , dx&=lim_{nrightarrow infty}sum_{i=o}^{n-1}f(x_{i})Delta x_{i}\
&=lim_{nrightarrow infty}frac{b-a}{n}sum_{i=0}^{n-1}lnleft(a+frac{(b-a)i}{n}right)\
&=lim_{nrightarrow infty}frac{b-a}{n}lnleft(prod_{i=0}^{n-1}left(a+frac{(b-a)i}{n}right)right)end{align}$$



And I'm stuck here, I don't know how to find this product, any hint would be appreciated , thank you !










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    up vote
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    I am being asked to calculate the integral $int_{a}^{b}ln x~dx$ using the definition of integral (i.e. expressing it as limit of Riemann sums)



    Here's what I did:
    Let's divide the interval $[a,b]$ into $n$ subintervals of the form $[x_{i},x_{i+1}]$.



    Let $x_i=a+frac{(b-a)i}{n}$ where $i=0,1,2...,n-1. $ be the start points of each subinterval such that $x_0=a$ and $x_n=b$



    Let $Delta x_i=x_i-x_{i-1}=frac{b-a}{n}$



    I defined the integral like this:



    $$begin{align}int_{a}^{b}ln x , dx&=lim_{nrightarrow infty}sum_{i=o}^{n-1}f(x_{i})Delta x_{i}\
    &=lim_{nrightarrow infty}frac{b-a}{n}sum_{i=0}^{n-1}lnleft(a+frac{(b-a)i}{n}right)\
    &=lim_{nrightarrow infty}frac{b-a}{n}lnleft(prod_{i=0}^{n-1}left(a+frac{(b-a)i}{n}right)right)end{align}$$



    And I'm stuck here, I don't know how to find this product, any hint would be appreciated , thank you !










    share|cite|improve this question


























      up vote
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      favorite
      2









      up vote
      2
      down vote

      favorite
      2






      2





      I am being asked to calculate the integral $int_{a}^{b}ln x~dx$ using the definition of integral (i.e. expressing it as limit of Riemann sums)



      Here's what I did:
      Let's divide the interval $[a,b]$ into $n$ subintervals of the form $[x_{i},x_{i+1}]$.



      Let $x_i=a+frac{(b-a)i}{n}$ where $i=0,1,2...,n-1. $ be the start points of each subinterval such that $x_0=a$ and $x_n=b$



      Let $Delta x_i=x_i-x_{i-1}=frac{b-a}{n}$



      I defined the integral like this:



      $$begin{align}int_{a}^{b}ln x , dx&=lim_{nrightarrow infty}sum_{i=o}^{n-1}f(x_{i})Delta x_{i}\
      &=lim_{nrightarrow infty}frac{b-a}{n}sum_{i=0}^{n-1}lnleft(a+frac{(b-a)i}{n}right)\
      &=lim_{nrightarrow infty}frac{b-a}{n}lnleft(prod_{i=0}^{n-1}left(a+frac{(b-a)i}{n}right)right)end{align}$$



      And I'm stuck here, I don't know how to find this product, any hint would be appreciated , thank you !










      share|cite|improve this question















      I am being asked to calculate the integral $int_{a}^{b}ln x~dx$ using the definition of integral (i.e. expressing it as limit of Riemann sums)



      Here's what I did:
      Let's divide the interval $[a,b]$ into $n$ subintervals of the form $[x_{i},x_{i+1}]$.



      Let $x_i=a+frac{(b-a)i}{n}$ where $i=0,1,2...,n-1. $ be the start points of each subinterval such that $x_0=a$ and $x_n=b$



      Let $Delta x_i=x_i-x_{i-1}=frac{b-a}{n}$



      I defined the integral like this:



      $$begin{align}int_{a}^{b}ln x , dx&=lim_{nrightarrow infty}sum_{i=o}^{n-1}f(x_{i})Delta x_{i}\
      &=lim_{nrightarrow infty}frac{b-a}{n}sum_{i=0}^{n-1}lnleft(a+frac{(b-a)i}{n}right)\
      &=lim_{nrightarrow infty}frac{b-a}{n}lnleft(prod_{i=0}^{n-1}left(a+frac{(b-a)i}{n}right)right)end{align}$$



      And I'm stuck here, I don't know how to find this product, any hint would be appreciated , thank you !







      real-analysis integration definite-integrals






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      edited Nov 17 at 14:57









      Masacroso

      12.3k41746




      12.3k41746










      asked Nov 17 at 14:18









      Maths Survivor

      441219




      441219






















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          Some hints:



          Let ${bover a}=:rho>1$, and choose an $Ngg1$. Then use the partition $a=x_0<x_1<ldots<x_N=b$ given by
          $$x_k:=a,rho^{k/N}qquad(0leq kleq N) .$$
          Then
          $$R_N:=sum_{k=0}^{N-1}log(x_k)(x_{k+1}-x_k)$$
          is an admissible Riemann sum for the given integral, i.e., we can be sure that $$lim_{Ntoinfty} R_N=int_a^blog x>dx .$$
          Now $R_N$ can be computed in an elementary way; you only need a formula for finite sums of the type $sum_{k=0}^{N-1} k, q^k$.






          share|cite|improve this answer





















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            up vote
            5
            down vote













            Some hints:



            Let ${bover a}=:rho>1$, and choose an $Ngg1$. Then use the partition $a=x_0<x_1<ldots<x_N=b$ given by
            $$x_k:=a,rho^{k/N}qquad(0leq kleq N) .$$
            Then
            $$R_N:=sum_{k=0}^{N-1}log(x_k)(x_{k+1}-x_k)$$
            is an admissible Riemann sum for the given integral, i.e., we can be sure that $$lim_{Ntoinfty} R_N=int_a^blog x>dx .$$
            Now $R_N$ can be computed in an elementary way; you only need a formula for finite sums of the type $sum_{k=0}^{N-1} k, q^k$.






            share|cite|improve this answer

























              up vote
              5
              down vote













              Some hints:



              Let ${bover a}=:rho>1$, and choose an $Ngg1$. Then use the partition $a=x_0<x_1<ldots<x_N=b$ given by
              $$x_k:=a,rho^{k/N}qquad(0leq kleq N) .$$
              Then
              $$R_N:=sum_{k=0}^{N-1}log(x_k)(x_{k+1}-x_k)$$
              is an admissible Riemann sum for the given integral, i.e., we can be sure that $$lim_{Ntoinfty} R_N=int_a^blog x>dx .$$
              Now $R_N$ can be computed in an elementary way; you only need a formula for finite sums of the type $sum_{k=0}^{N-1} k, q^k$.






              share|cite|improve this answer























                up vote
                5
                down vote










                up vote
                5
                down vote









                Some hints:



                Let ${bover a}=:rho>1$, and choose an $Ngg1$. Then use the partition $a=x_0<x_1<ldots<x_N=b$ given by
                $$x_k:=a,rho^{k/N}qquad(0leq kleq N) .$$
                Then
                $$R_N:=sum_{k=0}^{N-1}log(x_k)(x_{k+1}-x_k)$$
                is an admissible Riemann sum for the given integral, i.e., we can be sure that $$lim_{Ntoinfty} R_N=int_a^blog x>dx .$$
                Now $R_N$ can be computed in an elementary way; you only need a formula for finite sums of the type $sum_{k=0}^{N-1} k, q^k$.






                share|cite|improve this answer












                Some hints:



                Let ${bover a}=:rho>1$, and choose an $Ngg1$. Then use the partition $a=x_0<x_1<ldots<x_N=b$ given by
                $$x_k:=a,rho^{k/N}qquad(0leq kleq N) .$$
                Then
                $$R_N:=sum_{k=0}^{N-1}log(x_k)(x_{k+1}-x_k)$$
                is an admissible Riemann sum for the given integral, i.e., we can be sure that $$lim_{Ntoinfty} R_N=int_a^blog x>dx .$$
                Now $R_N$ can be computed in an elementary way; you only need a formula for finite sums of the type $sum_{k=0}^{N-1} k, q^k$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 17 at 14:45









                Christian Blatter

                171k7111325




                171k7111325






























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