Reimann Sum to Definite Integral: finding the limits

[theschaef]

Ok, heres the question... its in a booklet of old final exams and I keep getting a different answer then the one that is shown in the book. The question is: find an integral that equals

lim {n to infinity} [ (1/n) sigma {k=1 to n} sqrt [ ln (1+k/n) ] ]

So i made the limits of integration "a" and "b"

(b-a)/n = 1/n
b-a = 1

and then I said

1 + (k/n) = a + (b-a/n)(k)
a = 1
So b = 2

Giving the integral:
integral {1 to 2} sqrt [ ln (1+x) ]

but the actually answer is : integral {0 to 1} sqrt [ ln (1+x) ]
So im not sure where I messed up in finding the limits of integration.

 

[stapel]

Did they say which summation type they were using? That is, are they using right-hand sums, left-hand sums, or mid-point sums?

You have the correct interval length, since the fixed-width interval for the Riemann sum is of the form "(b - a)/n", and their sum uses "1/n", meaning that b - a = 1.

But note that, since the argument of your function is "1 + x", then only the "k/n" is the k-th x-value. For k = 1, the first x-value, x₁, is 1/n. Using the right-hand endpoint, and noting that each interval has a length of 1/n, this means that the first interval started at 1/n - 1/n = 0, and that zero is the value of "a" -- assuming they're using right-hand endpoints.

Eliz.

 

[theschaef]

The question simply states:

Which of the following integrals equals ... (the integral that i posted)
and then there are several choices.. the answer that i came up with is one of the choices but it is not labelled as the correct one.

So does that mean that the answer in the book is wrong? Or does it mean that both answers are possible? What is the difference between right-hand sums, left-hand sums and mid-point sums? This is an exam that is a few years old so maybe this has been taken out of the course..

 

[pka]

No the text is correct, here is why.
\(\displaystyle \Delta x = \frac{1}{n}\), you saw that correctly.
The Riemann points are \(\displaystyle x_k = a + k\Delta x\).
Thus for k=1 we should have \(\displaystyle f(x_1 ) = f(a + \Delta x) = \sqrt {\ln (1 + a + 1/n)}.\)
So for the given sum to be correct we must have a=0.

 

[stapel]

So does that mean that the answer in the book is wrong?

You were shown how the book's answer was correct, assuming that (as is customary) the Riemann sums were using the right-hand interval endpoints for the x-values. How would this "mean that the answer in the book is wrong"...?

Eliz.