# calc help

consider the function f(x)= x^2/4 -6
Rn is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval.

Calculate Rn for f(x)= x^2/4 -6 on the interval [0,4] and write your answer as a function of n without any summation signs.
Rn= ?

The k-th term in the summation is:

[(s*k)^2/4 - 6]*s

s is the step between two terms and is 4/n. k ranges from 1 to n. n is the number of parts in which you have divided the interval [0,4]. So, if you take n = 1 then s = 4 and you are then only evaluating the function x = 4, which is the right hand endpoint of the interval.

If you n = 2 then s = 2 and n ranges from 1 to 2. The function is ten evaluated at 1*s and 2*s, i.e. at 2 and 4 and the values are multiplied by 2.

To evaluate the summation we write:

[(k*4/n)^2/4 - 6]*4/n

= k^2 16/n^3 - 24/n

The last term is - 24/n. If we sum this from k = 1 to n then that amounts to mulitplying by n, so that yields -24.

To evaluate te first term we must calculate:

Sum from k=1 to n of k^2.

You can look up the formula for that (it's a third degree polynomial in n), but it is more fun to derive this yourself.

You can use the formula for the geometric series:

Sum from k=0 to n of a^k =

[1-a^(n+1)]/[1-a]

Differentiate both sides w.r.t. a:

Sum from k=1 to n of ka^(k-1) =

- (n+1)a^(n)/(1-a) + [1-a^(n+1)]/[1-a]^2

If you take the limit a-->1 on both sides you get the formula for the sum of k from 1 to n. We want the sum of k^2. If you differentiate again w.r.t. a a factor (k-1) comes down. That's not what we want, we want a factor k. So, you first multiply both sides by a and then you differentiate w.r.t. a. Then you take the limit a -->1 on both sides.

A faster way to calculate the summation is as follows.

Sum from k=0 to n of a^k =

[1-a^(n+1)]/[1-a]

Substitute a = exp[x] in here:

Sum from k=0 to n of exp[kx] =

[exp[(n+1)x] - 1]/[exp(x) - 1]

Expand both sides in powers of x. You can see that the coefficient of x^2 yields 1/2 times the desired summation.

To find the series expansion of the function

[exp[(n+1)x] - 1]/[exp(x) - 1]

you equate it to an unknown series:

c_0 + c_1x + c_2 x^2 + ...

This yields:

[c_0 + c_1x + c_2 x^2 + ...]*

[x + x^2/2 + x^3/6 + ...] =

(n+1)x + (n+1)^2x^2/2 + (n+1)x^3/6 + ...

And you find that:

c_0 = n+1

c_1 = 1/2 n(n+1)

c_2 = 1/6 n(n+1/2)(n+1)

The summation is 2c_2, so:

Sum from k = 1 to n of k^2 =

1/3 n(n+1/2)(n+1)

This means that the Riemann sum of the first term is:

16/3 n(n+1/2)(n+1)/n^3

1. 👍 0
2. 👎 0
3. 👁 235

## Similar Questions

1. ### Calculus

Evaluate the Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval. Give three decimal places in your answer and explain, using a graph of

asked by nan on March 10, 2016
2. ### calculus

consider the function f(x)= x^2/4 -6 Rn is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval. Calculate Rn for f(x)= x^2/4 -6 on the interval [0,4] and write your answer as a

asked by amanda on November 30, 2006
3. ### Calc 2

Can you give me the step by step instructions on how to do this problem? I'm having difficulty understanding Riemann Sum. Let f(x) = 2/x a. Compute the Riemann sum for R4 using 4 subintervals and right endpoints for the function

asked by Bae on May 2, 2014
4. ### Calculus

Evaluate the Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval and please explain, using a graph of f(x), what the Riemann sum

asked by Adrianna on March 11, 2016
5. ### Calculus

(a) Find the Riemann sum for f(x) = 7 sin x, 0 ≤ x ≤ 3π/2, with six terms, taking the sample points to be right endpoints. (Round your answers to six decimal places.) I got 3.887250 as an answer, but that's not right. (b)

asked by John on November 16, 2011
6. ### Calculus, Riemann

IF ANYONE CAN PLEAAAASEEEE HELP ME Use the Riemann sum definition of the definite integral to show that b 3 ¡ò x©÷dx - b©ø -a©ø a Here, for simplicity, we assume b > a > 0. Be sure to clearly specify your mesh/grid xi, ,

asked by Ellen D on May 7, 2014
7. ### calculus

Consider the given function. f(x) = 3 sin(6x) text(, ) 0

asked by Justin on January 18, 2008
8. ### calculus

Consider the given function. f(x) = 3 sin(6x) text(, ) 0

asked by Justin on January 18, 2008
9. ### calc

If f(x) = 3x^2 − 2x, 0 ≤ x ≤ 3, evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints.

asked by sara on April 16, 2015
10. ### Calculus

The Riemann sum s for f(x)=4x^2, 0

asked by Alice on April 19, 2016

More Similar Questions