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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.