• Support PF! Buy your school textbooks, materials and every day products Here!

Approximating integral using riemann sums

  • Thread starter missavvy
  • Start date
  • #1
82
0

Homework Statement


f: [0,1] -> Reals, f(x) = 3-x2
P={0,1/2,1}

Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error.


Homework Equations





The Attempt at a Solution


So I tried finding the upper Riemann sum
Spf = 3(1/4) + 2(1/2) = 19/8
and the lower:
spf = 3(1/2) + (3-1/4)(1/2) = 23/8

Is this correct? If so, how do I approximate the definate integral? Do I just pick the upper or lower sum and use that as my approximation? And how would I find the error?

Any help is appreciated :)
 

Answers and Replies

  • #2
196
0
The most accurate approximation would be to average the RHS and LHS. The error would be the absolute value of the difference between the RHS and LHS, or another way of looking at it is that its the absolute value of (f(b)-f(a))Δt, either way they should be non-negative.

As for the sums themselves, they way you wrote it a little confusing, but all you're doing is creating rectangles either above or below f(x) depending on the direction of the sum, that is left or right, and whether it increasing or decreasing.

Hope this helps a bit.
 
  • #3
82
0
Ok.. is Δt the difference between the x_j and x_j-1?

I added the upper and lower sums and divided by 2 and got 2.625 which is really close to the actual value of 2.6666...7.
For the error, I took the absolute value:
|19/8-23/8| = 0.5
So then 0.5 is the approximation error.. ?
 
  • #4
9
0
hi, I just thought I should point out that your upper riemann sum is less than your lower riemann sum.

I'm sure you're well aware that for any parition P, we have that Spf >= spf.

I suggest redrawing your graph making sure that you have chosen the appropriate height/width for the rectangles. good luck
 

Related Threads on Approximating integral using riemann sums

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
5K
Replies
1
Views
3K
  • Last Post
Replies
0
Views
955
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
7
Views
13K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
21
Views
2K
Top