# What am I doing wrong?

• Dec 6th 2012, 03:09 PM
Nervous
What am I doing wrong?
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...

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

$\displaystyle f(x) = 2x$

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

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

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

$\displaystyle \frac{2}{k^2}\sum_{j=1}^{k}j$
• Dec 6th 2012, 03:17 PM
Plato
Re: What am I doing wrong?
Quote:

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...
$\displaystyle \int f(x) dx = \lim_{k\to\infty}\sum_{j=1}^{k}(f(x_j) \Delta x)$
$\displaystyle f(x) = 2x$
$\displaystyle \Delta x = \frac{2 - 1}{k} = \frac{1}{k}$
$\displaystyle \color{red}x_j = \frac{1}{k} j = \frac{j}{k}$

It should be $\displaystyle x_j =1+ \frac{1}{k} j$
• Dec 6th 2012, 03:18 PM
Nervous
Re: What am I doing wrong?
continued...

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

And obviously...

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

So what am I doing wrong?
• Dec 6th 2012, 03:19 PM
Nervous
Re: What am I doing wrong?

I had to go to a second post because this tablet wouldn't let me scroll down...
• Dec 6th 2012, 03:25 PM
Nervous
Re: What am I doing wrong?
How did you get that?
• Dec 6th 2012, 03:44 PM
Plato
Re: What am I doing wrong?
Quote:

Originally Posted by Nervous
How did you get that?

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

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

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

etc.
• Dec 6th 2012, 04:45 PM
HallsofIvy
Re: What am I doing wrong?
Quote:

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!
• Dec 6th 2012, 05:24 PM
Plato
Re: What am I doing wrong?
Quote:

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.
• Dec 6th 2012, 05:40 PM
MarkFL
Re: What am I doing wrong?
Hey, that's in my neck of the woods! Was the school on State Road 16?
• Dec 6th 2012, 05:46 PM
Plato
Re: What am I doing wrong?
Quote:

Originally Posted by MarkFL2
Hey, that's in my neck of the woods! Was the school on State Road 16?

Not in my day.
• Dec 6th 2012, 05:50 PM
Plato
Re: What am I doing wrong?
Quote:

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.
• Dec 6th 2012, 06:07 PM
MarkFL
Re: What am I doing wrong?
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.