What am I doing wrong?

**Nervous** · December 6th 2012, 03:09 PM

Integrate f(x)=2x. Lower limit is 1 and upper limit is 2.

Using the shortcuts, it's easy for me to find F(2) - F(1) = 3.

However, I was asked to use the limit of Reinmann sums in order to solve the problem...

$\int f(x) dx = \lim_{k\to\infty}\sum_{j=1}^{k}(f(x_j) \Delta x)$

$f(x) = 2x$

$\Delta x = \frac{2 - 1}{k} = \frac{1}{k}$

$x_j = \frac{1}{k} j = \frac{j}{k}$

$\sum_{j=1}^{k} 2 \frac{j}{k} (\frac{1}{k})$

$\frac{2}{k^2}\sum_{j=1}^{k}j$

**Plato** · December 6th 2012, 03:17 PM

Originally Posted by Nervous

Integrate f(x)=2x. Lower limit is 1 and upper limit is 2.
Using the shortcuts, it's easy for me to find F(2) - F(1) = 3.
However, I was asked to use the limit of Reinmann sums in order to solve the problem...
$\int f(x) dx = \lim_{k\to\infty}\sum_{j=1}^{k}(f(x_j) \Delta x)$
$f(x) = 2x$
$\Delta x = \frac{2 - 1}{k} = \frac{1}{k}$
$\color{red}x_j = \frac{1}{k} j = \frac{j}{k}$

This is your mistake.

It should be $x_j =1+ \frac{1}{k} j$

**Nervous** · December 6th 2012, 03:18 PM

continued...

$\frac{2}{k^2} \frac{k^2 + k}{2} = \frac {2k^2 + 2k}{2k^2} = 1 + \frac {1}{k}$

And obviously...

$\lim_{k\to\infty} 1 + \frac {1}{k} = 1$

So what am I doing wrong?

**Nervous** · December 6th 2012, 03:19 PM

Thanks, just saw your post.

I had to go to a second post because this tablet wouldn't let me scroll down...

**Nervous** · December 6th 2012, 03:25 PM

How did you get that?

**Plato** · December 6th 2012, 03:44 PM

Originally Posted by Nervous

How did you get that?

To answer your question quite literally: I have been doing Riemann Sums since 1966.

On $[a,b],~\Delta x=\frac{b-a}{n}$ :
left-hand sums need: $x_j=a+(j-1)\Delta X,~j=1,2,\cdots,n$

right-hand sums need: $x_j=a+(j)\Delta X,~j=1,2,\cdots,n$

etc.

**HallsofIvy** · December 6th 2012, 04:45 PM

Originally Posted by Plato

To answer your question quite literally: I have been doing Riemann Sums since 1966.

My God! You are almost as old as I am!

**Plato** · December 6th 2012, 05:24 PM

Originally Posted by HallsofIvy

My God! You are almost as old as I am!

I am probably older than you. I am in the class of 1964, Millsaps College. I taught two years of high school in Green Cove Springs, FL.

**MarkFL** · December 6th 2012, 05:40 PM

Hey, that's in my neck of the woods! Was the school on State Road 16?

**Plato** · December 6th 2012, 05:46 PM

Originally Posted by MarkFL2

Hey, that's in my neck of the woods! Was the school on State Road 16?

Not in my day.

**Plato** · December 6th 2012, 05:50 PM

Originally Posted by Plato

Not in my day.

But we had Friday night dinner at Pappies in Saint Augustine.
Is it still there? I recall the huge tumblers of iced tea.

**MarkFL** · December 6th 2012, 06:07 PM

I think I know the place you are referring to. There used to be a seafood restaurant near the S.R. 207 and I-95 interchange by that name, but it has since become a tobacco shop I believe. The name is still the same though.

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