M: Solve Riemann Sum Problem Homework

[SYoungblood]

Homework Statement

[/B]
Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.

Homework Equations

f(x) = x²+ 1, over the interval [0,1]. This is problem number such-and-such from a well-known calculus textbook, not anything that is on an exam that I know of. The limit of these sums approaches infinity.

I'll call the Riemann Sum S_p, the length (x value) of the rectangles ∆x, and the height of the rectangles (y value) c_k.

The Attempt at a Solution

[/B]
∆x = 1/n, good to go.

c_k = a + k[(b - a)/n] = 0 + k (1/n) = k/n

S_p = Σ(k = 1, n) k/n * 1/n + 1

S_p = 1/n² Σ(k = 1, n) k + 1

And here is where I ran out of juice.

Again, and help is appreciated.

SY

 

[LCKurtz]

Homework Statement

[/B]
Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.

Homework Equations

f(x) = x²+ 1, over the interval [0,1]. This is problem number such-and-such from a well-known calculus textbook, not anything that is on an exam that I know of. The limit of these sums approaches infinity.

I'll call the Riemann Sum S_p, the length (x value) of the rectangles ∆x, and the height of the rectangles (y value) c_k.

The Attempt at a Solution

[/B]
∆x = 1/n, good to go.

c_k = a + k[(b - a)/n] = 0 + k (1/n) = k/n

S_p = Σ(k = 1, n) k/n * 1/n + 1

Since $f(x) = x^2 +1$ and $c_k = \frac k n$, then $f(c_k) = \frac {k^2} {n^2}+1$ so you should be calculating$$
\sum_{k=1}^n(\frac {k^2} {n^2}+1)\frac 1 n$$

You will need the formula for the sum of squares of the first n integers, which you can look up.

 

[SYoungblood]

I believe I just pulled a first-year calculus moron maneuver. Instead of k, it should be k²? And each term would then proceed to be determined using sigma notation for the degree of the individual term?

Here, that would be (as I see it),

S_p = 1/n² *
∑k=1n k² + ∑k=1n 1
= 1/n<sup>2 * [n(n+1)(2n+1)/6] + (n)-- my apologies for this being superscripted, I am having an issue w/ the formatting

Here, am I correct in understanding with the first sigma term, that reduces to n * 1/n2?

If I am, them we have 1/n * [(2n3+ n2+ 2n + n)/6]

= (2n3+ n2+3n)/6n

= (2n2+n + 3)/6

As I understand it, if i substitute the limit for n, infinity, then I am left with an area of 2/6 = 1/3 sp units. That doesn't pass the sniff test. What did I do wrong?

Thank you for your time.</sup>

 

[SammyS]

I believe I just pulled a first-year calculus moron maneuver. Instead of k, it should be k²? And each term would then proceed to be determined using sigma notation for the degree of the individual term?

Here, that would be (as I see it),

S_p = 1/n² *
∑k=1n k² + ∑k=1n 1
= 1/n<sup>2 * [n(n+1)(2n+1)/6] + (n)-- my apologies for this being superscripted, I am having an issue w/ the formatting

Here, am I correct in understanding with the first sigma term, that reduces to n * 1/n2?

If I am, them we have 1/n * [(2n3+ n2+ 2n + n)/6]

= (2n3+ n2+3n)/6n

= (2n2+n + 3)/6

As I understand it, if i substitute the limit for n, infinity, then I am left with an area of 2/6 = 1/3 sp units. That doesn't pass the sniff test. What did I do wrong?

Thank you for your time.</sup>

Read post #2.