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- Apr 10th 2008, 11:18 AMski4life912
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- Apr 10th 2008, 11:29 AMski4life912
Come to think of it maybe some kind of parabola or something where it changes direction and areas cancel out?

- Apr 10th 2008, 11:33 AMo_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.

- Apr 10th 2008, 11:47 AMski4life912
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.

- Apr 10th 2008, 12:00 PMo_O
Any linear equation would work really. For example, consider $\displaystyle y = -x + 3$ over the interval [-1, 2].

Length of each subinterval:

$\displaystyle \Delta x = \frac{2 - (-1)}{n} = \frac{3}{n}$

Consider your partition with left sample points:

$\displaystyle x_{0} = - 1 \:\: < \:\: x_{1} = -1 + \frac{3}{n} \:\: < \:\: x_{2} = -1 + 2\cdot \frac{3}{n} \:\: < \:\: ... \:\: < $ $\displaystyle {\color{blue}x_{i} = -1 + \frac{3i}{n}} \:\: < \:\: ... \:\: < \:\: x_{n} = -1 + \frac{3n}{n} = 2$

So your Riemann sum:

$\displaystyle \lim_{n \to \infty}\sum_{i = 1}^{n} f\left(x_{i}^{*}\right) \Delta x$

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

etc. etc.