Approximate an integral with left, right, or midpoint rectangles. The page traces how the inputs become riemann estimate before rounding.
It builds the definition of integration and gives controllable approximations from tabulated or computed values.
Review Function f(x) inside the Riemann Sum relation. Hold Lower bound steady while testing Upper bound. A rough Riemann estimate gives Riemann Sum an independent scale check. Keep the revised Riemann Sum inputs beside their own Riemann estimate.
Left and right sums can systematically under- or overestimate monotone functions; too few rectangles miss curvature.
Divide the interval evenly, select the stated point in each subinterval, multiply height by width, and add.
For x² on 0 to 3, midpoint rectangles approach the exact integral 9 as n grows. This Riemann Sum example can be compared with parabolic panels.
Link Function f(x) to its Riemann Sum role. Link Lower bound to its Riemann Sum role. The retained Riemann Sum formula identifies the Riemann Sum model.
A Riemann sum multiplies sampled heights by equal widths and adds the rectangle contributions. Riemann Sum can be checked against trapezoids.
This Riemann Sum calculation is based on Function f(x), Lower bound, Upper bound, Subinterval count, and Sample point. Use scale as a reasonableness test. Estimate the likely sign and order of magnitude from the graph or dominant term, then compare that expectation with the computed value before copying it into later work.
For this riemann sum result, if the result will feed another calculation, save several unrounded digits and the settings that produced them. Round only when presenting the final quantity, after checking that greater resolution does not move it materially.
A hand-worked reference for Riemann Sum need not duplicate the original numbers. Select an uncomplicated Function f(x), retain Lower bound, and estimate Riemann estimate. The reference gives the original Riemann Sum answer a meaningful scale check.
Midpoint often improves on one-sided rectangles.
Yes for signed integrands.
Under suitable conditions, the sum approaches the integral.