What are the requirements of a function so that the left Riemann sum equals the right Riemann sum?

Question

My homework question in particular specifies over an interval of [0,1], the function is negative, and the function is decreasing.

Answer 1

Given: $\Delta x = \frac{b - a}{n}$, when no limits are involved we can expand the left and right Riemann sums as follows:

\begin{align*} R_L &= \sum_{k = 0}^{n -1} f(a + k\Delta x) \Delta x = f(a) +\sum_{k = 1}^{n -1} f(a + k\Delta x) \Delta x \\ R_R &= \sum_{k = 1}^{n} f(a + k\Delta x) \Delta x = f(a+ n \Delta x) + \sum_{k = 1}^{n - 1} f(a + k\Delta x) \Delta x \\ \end{align*}

If $R_L = R_R$, then their difference should be zero:

\begin{align*} R_L - R_R &= 0\\ \left(f(a) +\sum_{k = 1}^{n -1} f(a + k\Delta x) \Delta x\right) - \left( f(a+ n \Delta x) + \sum_{k = 1}^{n - 1} f(a + k\Delta x) \Delta x\right) &= 0 \\ f(a) - f(a+ n \Delta x) &= 0 \\ \end{align*}

Therefore, the left and right Riemann sums are the same, without limits, when the following holds true:

$$f(a) = f(a + n \Delta x)$$

When limits are involved, the left Riemann sum approaches the right Riemann sum, as $n \to \infty$.

\begin{align*} \lim_{n \to \infty} R_L &= \lim_{n \to \infty} R_R \\ \end{align*}