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.
A Riemann sum multiplies sampled heights by equal widths and adds the rectangle contributions. Riemann Sum can be checked against trapezoids.
The meaning of riemann estimate follows from the relationship, so a different setup can produce a valid answer to a different question.
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.
Left and right sums can systematically under- or overestimate monotone functions; too few rectangles miss curvature.
Changing only one field helps distinguish a data-entry problem from the intended behavior of riemann sum.
Divide the interval evenly, select the stated point in each subinterval, multiply height by width, and add.
Transfer the source entries one at a time under Function f(x), Lower bound, Upper bound, Subinterval count and Sample point, keeping any units visible in your notes. This page can validate numerical ranges, but only the reader can confirm that each entry represents the intended quantity.
Increasing n usually improves resolution but also increases work. Trying a nearby input should follow this pattern; otherwise inspect the field assignment.
Trapezoidal and Simpson rules combine samples to achieve different error behavior. That neighboring measure needs its own setup rather than a relabeled answer.
Attach units and the name riemann estimate whenever the answer leaves this page. A rounded value is suitable for presentation, while retained working digits are safer for a dependent calculation.
Copying only the decimal discards the setup. Pair the riemann estimate with Function f(x), Lower bound, Upper bound, Subinterval count and Sample point and the unit convention so another reader can reconstruct its meaning.
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.
Midpoint often improves on one-sided rectangles.
Yes for signed integrands.
Under suitable conditions, the sum approaches the integral.