Riemann Sums

The Riemann sum approximates the area between the graph of a function and the \(x\)-axis as a sum of areas of rectangles. Different methods of selecting the heights of the rectangles yield slightly different approximations; observe these differences and see how the sum changes as the number of rectangles used in the calculation changes. Compare the Riemann sum to estimates obtained using the trapezoidal rule and Simpson's rule.

Choose a function \(f(x)\) from a pull-down menu at lower left, and a method of selecting the heights of rectangles in a Riemann sum from the column of check boxes at bottom. The choices are "Min" (use the minimum value in each interval), "Max" (use the maximum value in each interval), the trapezoidal rule, Simpson's rule, and "Evaluation point". If "Evaluation point" is selected, a slider appears, on which to select the point in the each interval used to evaluate \(f(x)\). Selecting \(0.0\) gives the left end point evaluation; selecting \(1.0\) gives the right end point evaluation.

The main graphing window shows the graph of the selected function in cyan, and the Riemann sum rectangles in yellow. If "Evaluation point" is selected, the point at which the evaluation is performed is marked in each interval.

A cursor rollover invokes crosshairs and a readout of the coordinates. Depressing the mousekey suppresses them.

At right is a slider controlling \(\Delta x\) and the number of intervals, \(n\). The value of the corresponding Riemann sum is marked on the graphing window at right, above the value of \(\Delta x\). A readout of the value of the sum is given below.