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Show that $f(x)=0,;0leq x<1/2,; f(x)=1,;1/2leq xleq 1$ is Riemann integrable over $[0,1]$ and find its...
$begingroup$
I want to prove that begin{align} f:[0&,1]to Bbb{R}\&xmapsto begin{cases}0,&0leq x<1/2,\\ 1,&1/2leq xleq 1. end{cases}end{align}
is Riemann integrable over $[0,1]$ and also, I want to find the value.
MY TRIAL
Let $epsilon>0,;x_1in [0,1/2)$ and $x_2in [1/2,1]$ such that $x_2-x_1<epsilon.$ The set
$$P={0,x_1,x_2,1 }$$
forms a partition of $[0,1].$ Define $Delta x_j=(x_j-x_{j-1})$ and $I_j=[x_{j-1}-x_j],;j=1,2,3.$ So, the upper and lower Darboux sums are given by
$$U(f,P)=sum^{3}_{j=1}M_j Delta x_j=1-x_1,;;text{where};;M_j=sup_{xin I_j}f(x),$$
$$L(f,P)=sum^{3}_{j=1}m_j Delta x_j=1-x_2.$$
Then,$$U(f,P)-L(f,P)=x_2-x_1<epsilon.$$
Hence, $f$ is Riemann integrable over $[0,1]$.
I have been able to show, as seen above, that $f$ is Riemann integrable over $[0,1]$ but not able to find the value of the integral. I know that the answer is $1/2$ but can anyone help out by showing how I could arrive at it?
real-analysis analysis riemann-integration
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
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add a comment |
$begingroup$
I want to prove that begin{align} f:[0&,1]to Bbb{R}\&xmapsto begin{cases}0,&0leq x<1/2,\\ 1,&1/2leq xleq 1. end{cases}end{align}
is Riemann integrable over $[0,1]$ and also, I want to find the value.
MY TRIAL
Let $epsilon>0,;x_1in [0,1/2)$ and $x_2in [1/2,1]$ such that $x_2-x_1<epsilon.$ The set
$$P={0,x_1,x_2,1 }$$
forms a partition of $[0,1].$ Define $Delta x_j=(x_j-x_{j-1})$ and $I_j=[x_{j-1}-x_j],;j=1,2,3.$ So, the upper and lower Darboux sums are given by
$$U(f,P)=sum^{3}_{j=1}M_j Delta x_j=1-x_1,;;text{where};;M_j=sup_{xin I_j}f(x),$$
$$L(f,P)=sum^{3}_{j=1}m_j Delta x_j=1-x_2.$$
Then,$$U(f,P)-L(f,P)=x_2-x_1<epsilon.$$
Hence, $f$ is Riemann integrable over $[0,1]$.
I have been able to show, as seen above, that $f$ is Riemann integrable over $[0,1]$ but not able to find the value of the integral. I know that the answer is $1/2$ but can anyone help out by showing how I could arrive at it?
real-analysis analysis riemann-integration
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
$endgroup$
add a comment |
$begingroup$
I want to prove that begin{align} f:[0&,1]to Bbb{R}\&xmapsto begin{cases}0,&0leq x<1/2,\\ 1,&1/2leq xleq 1. end{cases}end{align}
is Riemann integrable over $[0,1]$ and also, I want to find the value.
MY TRIAL
Let $epsilon>0,;x_1in [0,1/2)$ and $x_2in [1/2,1]$ such that $x_2-x_1<epsilon.$ The set
$$P={0,x_1,x_2,1 }$$
forms a partition of $[0,1].$ Define $Delta x_j=(x_j-x_{j-1})$ and $I_j=[x_{j-1}-x_j],;j=1,2,3.$ So, the upper and lower Darboux sums are given by
$$U(f,P)=sum^{3}_{j=1}M_j Delta x_j=1-x_1,;;text{where};;M_j=sup_{xin I_j}f(x),$$
$$L(f,P)=sum^{3}_{j=1}m_j Delta x_j=1-x_2.$$
Then,$$U(f,P)-L(f,P)=x_2-x_1<epsilon.$$
Hence, $f$ is Riemann integrable over $[0,1]$.
I have been able to show, as seen above, that $f$ is Riemann integrable over $[0,1]$ but not able to find the value of the integral. I know that the answer is $1/2$ but can anyone help out by showing how I could arrive at it?
real-analysis analysis riemann-integration
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
$endgroup$
I want to prove that begin{align} f:[0&,1]to Bbb{R}\&xmapsto begin{cases}0,&0leq x<1/2,\\ 1,&1/2leq xleq 1. end{cases}end{align}
is Riemann integrable over $[0,1]$ and also, I want to find the value.
MY TRIAL
Let $epsilon>0,;x_1in [0,1/2)$ and $x_2in [1/2,1]$ such that $x_2-x_1<epsilon.$ The set
$$P={0,x_1,x_2,1 }$$
forms a partition of $[0,1].$ Define $Delta x_j=(x_j-x_{j-1})$ and $I_j=[x_{j-1}-x_j],;j=1,2,3.$ So, the upper and lower Darboux sums are given by
$$U(f,P)=sum^{3}_{j=1}M_j Delta x_j=1-x_1,;;text{where};;M_j=sup_{xin I_j}f(x),$$
$$L(f,P)=sum^{3}_{j=1}m_j Delta x_j=1-x_2.$$
Then,$$U(f,P)-L(f,P)=x_2-x_1<epsilon.$$
Hence, $f$ is Riemann integrable over $[0,1]$.
I have been able to show, as seen above, that $f$ is Riemann integrable over $[0,1]$ but not able to find the value of the integral. I know that the answer is $1/2$ but can anyone help out by showing how I could arrive at it?
real-analysis analysis riemann-integration
real-analysis analysis riemann-integration
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
asked Jan 22 at 15:50
Omojola MichealOmojola Micheal
1,969324
1,969324
add a comment |
add a comment |
1 Answer
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For each $ninmathbb N$, let $P_n=left{0,frac12,frac12+frac1{2n},1right}$. Compute $U(f,P_n)$ and $L(f,P_n)$ for each natural $n$. What do you get?
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
$endgroup$
$begingroup$
How did you think of that partition? No answer required! That's unthoughtful of me! Thanks, (+1)
$endgroup$
– Omojola Micheal
Jan 22 at 16:02
$begingroup$
That jump at $frac12$ is what made me think of that partition.
$endgroup$
– José Carlos Santos
Jan 22 at 16:06
$begingroup$
Thanks a lot! I should have thought of that, too! Anyway, thank you very much!
$endgroup$
– Omojola Micheal
Jan 22 at 16:07
add a comment |
Your Answer
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1 Answer
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$begingroup$
For each $ninmathbb N$, let $P_n=left{0,frac12,frac12+frac1{2n},1right}$. Compute $U(f,P_n)$ and $L(f,P_n)$ for each natural $n$. What do you get?
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
$endgroup$
$begingroup$
How did you think of that partition? No answer required! That's unthoughtful of me! Thanks, (+1)
$endgroup$
– Omojola Micheal
Jan 22 at 16:02
$begingroup$
That jump at $frac12$ is what made me think of that partition.
$endgroup$
– José Carlos Santos
Jan 22 at 16:06
$begingroup$
Thanks a lot! I should have thought of that, too! Anyway, thank you very much!
$endgroup$
– Omojola Micheal
Jan 22 at 16:07
add a comment |
$begingroup$
For each $ninmathbb N$, let $P_n=left{0,frac12,frac12+frac1{2n},1right}$. Compute $U(f,P_n)$ and $L(f,P_n)$ for each natural $n$. What do you get?
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
$endgroup$
$begingroup$
How did you think of that partition? No answer required! That's unthoughtful of me! Thanks, (+1)
$endgroup$
– Omojola Micheal
Jan 22 at 16:02
$begingroup$
That jump at $frac12$ is what made me think of that partition.
$endgroup$
– José Carlos Santos
Jan 22 at 16:06
$begingroup$
Thanks a lot! I should have thought of that, too! Anyway, thank you very much!
$endgroup$
– Omojola Micheal
Jan 22 at 16:07
add a comment |
$begingroup$
For each $ninmathbb N$, let $P_n=left{0,frac12,frac12+frac1{2n},1right}$. Compute $U(f,P_n)$ and $L(f,P_n)$ for each natural $n$. What do you get?
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
$endgroup$
For each $ninmathbb N$, let $P_n=left{0,frac12,frac12+frac1{2n},1right}$. Compute $U(f,P_n)$ and $L(f,P_n)$ for each natural $n$. What do you get?
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
answered Jan 22 at 15:55
José Carlos SantosJosé Carlos Santos
166k22132235
166k22132235
$begingroup$
How did you think of that partition? No answer required! That's unthoughtful of me! Thanks, (+1)
$endgroup$
– Omojola Micheal
Jan 22 at 16:02
$begingroup$
That jump at $frac12$ is what made me think of that partition.
$endgroup$
– José Carlos Santos
Jan 22 at 16:06
$begingroup$
Thanks a lot! I should have thought of that, too! Anyway, thank you very much!
$endgroup$
– Omojola Micheal
Jan 22 at 16:07
add a comment |
$begingroup$
How did you think of that partition? No answer required! That's unthoughtful of me! Thanks, (+1)
$endgroup$
– Omojola Micheal
Jan 22 at 16:02
$begingroup$
That jump at $frac12$ is what made me think of that partition.
$endgroup$
– José Carlos Santos
Jan 22 at 16:06
$begingroup$
Thanks a lot! I should have thought of that, too! Anyway, thank you very much!
$endgroup$
– Omojola Micheal
Jan 22 at 16:07
$begingroup$
How did you think of that partition? No answer required! That's unthoughtful of me! Thanks, (+1)
$endgroup$
– Omojola Micheal
Jan 22 at 16:02
$begingroup$
How did you think of that partition? No answer required! That's unthoughtful of me! Thanks, (+1)
$endgroup$
– Omojola Micheal
Jan 22 at 16:02
$begingroup$
That jump at $frac12$ is what made me think of that partition.
$endgroup$
– José Carlos Santos
Jan 22 at 16:06
$begingroup$
That jump at $frac12$ is what made me think of that partition.
$endgroup$
– José Carlos Santos
Jan 22 at 16:06
$begingroup$
Thanks a lot! I should have thought of that, too! Anyway, thank you very much!
$endgroup$
– Omojola Micheal
Jan 22 at 16:07
$begingroup$
Thanks a lot! I should have thought of that, too! Anyway, thank you very much!
$endgroup$
– Omojola Micheal
Jan 22 at 16:07
add a comment |
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