# Riemann Sums

• April 10th 2008, 12:18 PM
ski4life912
.. ..
• April 10th 2008, 12:29 PM
ski4life912
Come to think of it maybe some kind of parabola or something where it changes direction and areas cancel out?
• April 10th 2008, 12:33 PM
o_O
Did you type the question correctly or am I missing something (Thinking)? There are several functions which are continuous over [-1,2] with partitions of that same interval used to calculate the areas under them.
• April 10th 2008, 12:47 PM
ski4life912
Well right, the general question was do they exist? if so give an example if not why not, at first I didn't think they existed but now im just thinking of examples that fit that.
• April 10th 2008, 01:00 PM
o_O
Any linear equation would work really. For example, consider $y = -x + 3$ over the interval [-1, 2].

Length of each subinterval:
$\Delta x = \frac{2 - (-1)}{n} = \frac{3}{n}$

Consider your partition with left sample points:
$x_{0} = - 1 \:\: < \:\: x_{1} = -1 + \frac{3}{n} \:\: < \:\: x_{2} = -1 + 2\cdot \frac{3}{n} \:\: < \:\: ... \:\: <$ ${\color{blue}x_{i} = -1 + \frac{3i}{n}} \:\: < \:\: ... \:\: < \:\: x_{n} = -1 + \frac{3n}{n} = 2$

So your Riemann sum:
$\lim_{n \to \infty}\sum_{i = 1}^{n} f\left(x_{i}^{*}\right) \Delta x$

$x_{i}^{*} = x_{i}$

etc. etc.