Write the definite integral for the summation:

Question

n
Lim Σ (1+ k/n)^2 * (1/n)
n->∞ k=1

1
A. ∫ x^2 dx
0

2
B. ∫ (x+1)^2 dx
1

2
C. ∫ x^2 dx
1

Thank you in advance for the help!

1
D. ∫ x^2 dx
2

Answer 1

I'm sorry if the problem looks a bit confusing. It was a little hard to type it.
The upper bound of sigma notation is n and the lower bound is k=1.

*And the upper and lower bound of choices A through D are the numbers that I accidentally placed right above and below each Choice letter.

Answer 2

Since the typing is most likely very confusing, here is the link with the pictures of the problems. 👍
docs.google.com/document/d/1ANn3kbGwHEal4-dyBMSt0q1t6bXKUZqbf6zEPxT48CA/edit?usp=sharing

Answer 3

Make sure to add the, h t t p s : / / before the link. I guess Jiskha doesn't allow links to be posted.

Answer 4

The problem of typing something like an integral is that this format
does not accept spaces, so we have to work around that
e.g.
B could be typed like this:
∫ (x+1)^2 dx , x goes from 1 to 2

Wolfram accepts that terminology.
www.wolframalpha.com/input/?i=%E2%88%AB+(x%2B1)%5E2+dx+,+x+goes+from+1+to+2

Ok, I will do that one
∫ (x+1)^2 dx , x goes from 1 to 2
= (1/3)(x+1)^3 | from 1 to 2
= (1/3)(3^3) - (1/3)2^3
= 27/3 - 8/3 = 19/3

for the others, you should be able to integrate polynomials of this type

Answer 5

@Reiny
So I have to integrate each of them? Or are you saying that B is the right option(or did you use that as an example/)

Answer 6

You know that the area under a curve (the integral) can be approximated by a Riemann sum of rectangles, right?

Now consider this sum for for some value of n. You have
Σ (1+ k/n)^2 * (1/n)
The height of each rectangle is 1 + k/n
the width is 1/n
So, you have divided an interval of width 1 into n strips.
You are using the right-endpoint of each subinterval (since k goes from 1..n)
Consider the case where n=3. Your sum is
[(1+1/3)^2 + (1+2/3)^2 + (1+3/3)^2 ] (1/3) ≈ ∫[1,2] x^2 dx