Find Area Under Curve y=x^3 from 0 to 1: Riemann Sum Limit

[QuarkCharmer]

Homework Statement

a.) Use definition 2 to find an expression for the area under the curve y=x^3 from 0 to 1 as a limit.
b.)Evaluate the (above) limit using the sum of the cubes of the n integers.

Homework Equations

$$(\frac{n(n+1)}{2})^{2}$$

The Attempt at a Solution

For part a.) I wrote my limit like this:
$$\lim_{n \to \infty} \Sigma_{i=1}^{n}(\frac{i}{n})^{3}\frac{1}{n}$$The "Definition 2" they have listed just says:
$$A=\lim_{n \to \infty} R_{n} = \lim_{n \to \infty}(f(x_{1})\Delta x + f(x_{2})\Delta x + . . . + f(x_{n})\Delta x)$$

Now for part b, I understand that the formula is the sum of all cubes and so on. So I am thinking that the limit should look like this?

$$\lim_{n \to \infty}(\frac{n(n+1)}{2})^{2}$$
That should handle the limit and the sum of cubes, now I need each one to multiply by delta x right? So that it comes out to:
$$\lim_{n \to \infty}(\frac{n(n+1)}{2})^{2}\frac{1}{n}$$
..because the integral is from 0 to 1, so $\Delta x = \frac{1-0}{n}$

But I am not sure how to write it in this manner and take the limit from here yet. Is this correct so far? I should just simplify the expression after the limit and then take the limit?

 

[Dick]

No, no. The sum of i^3 for i=0 to n is (n(n+1)/2))^2. That's only a PART of the sum of (i/n)^3*(1/n) for i=0 to n. Do some algebra in that sum to separate out the i^3 part and then substitute your summation formula and take the limit.

 

[QuarkCharmer]

I understand now. Sorry, this is the first time I have ever actually used a summation, other than just knowing what it is.

I checked with the definite integral and this is correct as far as I can tell.
I'm going to go learn the rules of summation and find some problems to do. Thanks, I didn't realize that this was how it worked.

 

[Dick]

Yes, the integral from x=0 to x=1 is 1/4. And I think you did it correctly through the limit.

 

[QuarkCharmer]

Thanks for the help!

 

[ScienceGeek24]

Hey can someone explain to me how did he jumped from lim n-> Ʃ i^3/n * 1/n to lim n->1/n^4 Ʃ i^3??

 

[Dick]

Hey can someone explain to me how did he jumped from lim n-> Ʃ i^3/n * 1/n to lim n->1/n^4 Ʃ i^3??

He didn't jump anywhere. He used a formula the sum of i^3 for i=1 to n.

 

[ScienceGeek24]

I see... but he committed a mistake in the evaluation at lim n-> Ʃ i^3/n * 1/n is not lim n-> Ʃ i^3/n * 1/n is lim n-> Ʃ i^3/n^3 * 1/n

 

[ScienceGeek24]

I was having a hard time finding how did he got that 1/n^4... LOL it was an ibvious mistake from part (a) even part (a) is wrong.

 

[ScienceGeek24]

deltax= 1-0/n=1/n than according to definition 2 the R endpoints formula is (a+ideltax)deltax so is i/n*1/n than substituting from x^3 we have (i/n)^3*1/n=i^3/n^3*1/n=i^3/n^4 which makes more sense. Now i just get rid of the 1/n^4 and put it in the other side of the summation and ta chan! problem solved LOL

 

[Dick]

I see... but he committed a mistake in the evaluation at lim n-> Ʃ i^3/n * 1/n is not lim n-> Ʃ i^3/n * 1/n is lim n-> Ʃ i^3/n^3 * 1/n

It's pretty clear he meant (i/n)^3 not i^3/n since the n^4 appears in the next line. Why are you resurrecting posts over to a year old to complain about notational mistakes?

 

[ScienceGeek24]

is not clear in part (a).

 

[Dick]

is not clear in part (a).

I'll admit the way it was written isn't clear. Using an extra parentheses would have helped.

 

[ScienceGeek24]

I'll admit the way it was written isn't clear. Using an extra parentheses would have helped.

Thank you.