Approximating integral using riemann sums

[missavvy]

Homework Statement

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

Find lower and upper Riemann sums, and approximate the definite 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 definite 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 :)

 

[armolinasf]

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.

 

[missavvy]

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.. ?

 

[a1010711]

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