Is Right hand, left hand, and middle riemann sum formula the same?

Question

Are the right hand, left hand, and middle Riemann sum formulas for $\int_{a}^{b}f(x)\,dx$ the same? $$\lim_{n\to\infty}\sum_{i=1}^{n}f(x_{i})\Delta x,$$ where $\Delta x =\frac{b-a}{n}$ and $x_i=a+i\Delta x$. Is this correct?

Answer 1

No, the formulas are not the same, but they are similar. Consider an interval $[a,b]$ and a partition of the interval $a = x_{1} < x_{2} < \dotsb < x_{n} = b$, where $\Delta x = \frac{b-a}{n}$ is the spacing between points. Then the formulas for the Riemann sums will change, depending on which point (right/left/middle) in the subinterval is chosen: \begin{align} \text{Right:}\quad \int_{a}^{b}f(x)\,dx &= \lim_{n\to\infty}\sum_{i=2}^{n}f(x_{i})\Delta x\\ \text{Left:}\quad \int_{a}^{b}f(x)\,dx &= \lim_{n\to\infty}\sum_{i=1}^{n-1}f(x_{i})\Delta x\\ \text{Middle:}\quad \int_{a}^{b}f(x)\,dx &= \lim_{n\to\infty}\sum_{i=1}^{n-1}f\left(\frac{x_{i+1}-x_{i}}{2}\right)\Delta x\\ &=\lim_{n\to\infty}\sum_{i=2}^{n}f\left(\frac{x_{i}-x_{i-1}}{2}\right)\Delta x \end{align}

It is easiest to see why this is the case if you sketch out a graph and label some points. You should be able to see why the indices on the sums change depending on whether you pick the right or left point on each sub-interval.