Single Variable Calculus, Chapter 5, 5.2, Section 5.2, Problem 2

Suppose that $f(x) = x^2 - 2x, 0 \leq x \leq 3$, find the Riemann sum with $n = 6$, taking the sample points to be right end points. What does Riemann sum represent?

With $n = 6$, we divide the interval $(0,3)$ into 6 rectangles with widths

$\displaystyle \Delta x = \frac{3-0}{6} = 0.5$ at $x = 0, 0.5, 1, 1.5, 2, 2.5$ and $3$.

Evaluating $f(x)$ on the right end points (starting from $x = 0.5$)


$\begin{array}{|c|c|} \hline\\ x & f(x) = x^2 - 2x \\ 0.5 & -0.75 \\ 1 & -1 \\ 1.5 & -0.75 \\ 2 & 0 \\ 2.5 & 1.25 \\ 3 & 3\\ \hline \end{array}$


Now, the total area of the rectangle is..

$0.5(-0.75 - 1 - 0.75 + 0 + 1.25 + 3) = 0.875$

The Riemann sum represents an estimate of the area between the curve and the $x$-axis. Although in some cases, some areas result to a negative value because some rectangles are located below the $x$-axis. With this, you have to take the absolute values of such areas to get the actual area.