# Thread: Upper and Lower Riemann Sums

1. ## Upper and Lower Riemann Sums

Compute the upper and lower Riemann sums and the integral from zero to 1 of f(x)dx, where the partition is

P= {0, 2/5, 1/2, 3/5, 1}

Thanks.

2. Originally Posted by taypez
Compute the upper and lower Riemann sums and the integral from zero to 1 of f(x)dx, where the partition is

P= {0, 2/5, 1/2, 3/5, 1}

Thanks.
What is f(x)?

3. f(x) = 1-x^2

4. Originally Posted by taypez
Compute the upper and lower Riemann sums and the integral from zero to 1 of f(x)dx, where the partition is

P= {0, 2/5, 1/2, 3/5, 1}

Thanks.
$\displaystyle f(x) = 1 - x^2$

f(x) is decreasing from 0 to 1. hence, for the given partition:

$\displaystyle U(P,f) = 1\cdot \frac{2}{5} + \frac{21}{25} \cdot \frac{1}{10} + \frac{3}{4} \cdot \frac{1}{10} + \frac{16}{25} \cdot \frac{2}{5} = \frac{163}{200}$

$\displaystyle L(P,f) = \frac{21}{25} \cdot \frac{2}{5} + \frac{3}{4} \cdot \frac{1}{10} + \frac{16}{25} \cdot \frac{1}{10} + 0 \cdot \frac{2}{5} = \frac{19}{40}$

5. Can you give me an example of a couple calculations on how you arrived at these numbers for U(f,P)? I came up with some different ones and I can't find my mistake.

Thanks.

6. Originally Posted by taypez
Can you give me an example of a couple calculations on how you arrived at these numbers for U(f,P)? I came up with some different ones and I can't find my mistake.

Thanks.
Because U(f,P) is calculated by taking the supremum (largest value) on each interval. So since the function was decreasing the largest value is the left-endpoint.