Math Insight

Applet: Area via a left Riemann sum

The area underneath the graph of $f(x)$ (blue curve in left panel) over the interval $[a,b]$ is calculated via a left Riemann sum. The left Riemann sum of $n$ subintervals is illustrated by the rectangles superimposed with the graph of $f$. The right panel shows the area of the rectangles $\hat{A}(x)$ from $a$ to $x$, plotted as a green curve. The area over the whole interval $[a,b]$ is the value $\hat{A}(b)$. To investigate the behavior of $\hat{A}$, you can move pink points along the curve and the tops of the rectangles. As you move the pink points, a rectangle is highlighted, and the calculation of its area is shown in the upper right corner. The area of each rectangle is the value of $f$ at its left endpoint times the subinterval width $\Delta x$. The running sum of the area, $\hat{A}(x)$, increases by the area of a rectangle when you move the pink points right one rectangle. If you check the “exact” box, the true area under the graph of $f$ is shaded in red at the left and the right panel displays a graph (in red) of the true area $A(x)$ under $f$ from $a$ to $x$. The true area through the right of the highlighted rectangle is calculated along with the error between the true area and the corresponding area calculated with the Riemann sum. The values of $A(x)$ and $\hat{A}(x)$ are areas under $f$ only for the case when $f(x) \ge 0$.

Applet file: area_left_riemann_sum.ggb