Question on Reimann Sums

[bbl]

I was reviewing past exams and this problem was given:


Is it right to say that the integral is [imath]\int_{0}^{10000} \sqrt{x} dx[/imath]? Or is my integral wrong?

 

[BigBeachBanana]

I was reviewing past exams and this problem was given:

View attachment 32424
Is it right to say that the integral is [imath]\int_{0}^{10000} \sqrt{x} dx[/imath]? Or is my integral wrong?

The lower bound should be 1 since the summation starts at 1, but everything else looks fine.

 

[bbl]

The lower bound should be 1 since the summation starts at 1, but everything else looks fine.

Thank you!

 

[Dr.Peterson]

The lower bound should be 1 since the summation starts at 1, but everything else looks fine.

I'm not so sure of that. But do we want an under- or over-estimate?

 

[BigBeachBanana]

I'm not so sure of that. But do we want an under- or over-estimate?

hm...I need to review my Riemann Sum definition and give this some thought.

 

[Cubist]

hm...I need to review my Riemann Sum definition and give this some thought.

This can be remembered by thinking about an extreme case like [imath]\sum_{i=1}^1\sqrt{i}[/imath] which equals 1.
This wouldn't make sense as [imath]\int_1^1\sqrt{x}\,dx[/imath] since this works out to 0.

 

[BigBeachBanana]

This can be remembered by thinking about an extreme case like [imath]\sum_{i=1}^1\sqrt{i}[/imath] which equals 1.
This wouldn't make sense as [imath]\int_1^1\sqrt{x}\,dx[/imath] since this works out to 0.

Good explanation.

 

[Dr.Peterson]

hm...I need to review my Riemann Sum definition and give this some thought.

What I would do is to sketch the graph for a smaller case, like the sum from 1 to 4, and think about what integral would correspond to the Riemann sum.

The other issue I hinted at is that there is not really one Riemann sum, but many. You could use a left Riemann sum, or a right Riemann sum, or others, and you'd get a slightly different estimate in each case. So the phrase "the Riemann sum" in the hint is misleading; but "use an integral" in the problem itself is appropriate!

 

[BigBeachBanana]

What I would do is to sketch the graph for a smaller case, like the sum from 1 to 4, and think about what integral would correspond to the Riemann sum.

Yes, it is clear from the smaller case.

The other issue I hinted at is that there is not really one Riemann sum, but many. You could use a left Riemann sum, or a right Riemann sum, or others, and you'd get a slightly different estimate in each case. So the phrase "the Riemann sum" in the hint is misleading; but "use an integral" in the problem itself is appropriate!

Of course, I should know this.

 

[Steven G]

To OP,
In my opinion you need to think this out on your own. Draw the y = sqrt(x) say going from 1 to 5. Does this area (use integration) approximate sqrt(1) + sqrt(2) + ... + sqrt(5)?
By looking and seeing is the way to learn math.