Trapezoidal Rule with sample values
For x² on 0 to 3, increasing n drives the estimate toward 9.
Math calculator
Trapezoidal Rule Calculator
Approximate a definite integral with straight-line panels. Beside trapezoidal estimate, the output shows the operation used to obtain it.
Trapezoidal Rule inputs
Build the expression
The mathematical idea behind Trapezoidal Rule
The trapezoidal rule averages neighboring endpoint heights and multiplies by panel width.
Checking Trapezoidal Rule away from the browser
It integrates tabulated data and smooth functions without derivative formulas. A related application of Trapezoidal Rule is rectangle sums.
Failure points in a Trapezoidal Rule setup
Strong curvature or discontinuities require more panels; straight chords bias convex and concave regions differently.
The response pattern for Trapezoidal Rule
Weight endpoints by one half, interior samples fully, and multiply their sum by h. Trapezoidal Rule also leads to Simpson comparison.
Reading numerical evidence carefully
Record Function f(x), Lower bound, Upper bound, and Subinterval count when saving the Trapezoidal Rule result. Keep function syntax explicit: write multiplication signs, balanced parentheses, and the intended variable. The restricted parser avoids arbitrary code, but it cannot infer omitted multiplication or decide which textbook convention an ambiguous expression intended.
For this trapezoidal rule result, reproduction requires more than copying the answer. Preserve the entered function, the active variable, and every endpoint or starting value so another calculation follows the same mathematical problem. Compare against a doubled panel count and watch whether the change shrinks at the expected rate.
Confirming the Trapezoidal Rule setup
Inspect the domain of Function f(x) before using Trapezoidal Rule. Keep exact Trapezoidal Rule work separate from Lower bound.
Test Trapezoidal Rule on a constant or linear function. Refine any numerical Trapezoidal Rule step and compare the approximation.
Before accepting Trapezoidal estimate, restore the Trapezoidal Rule inputs Function f(x) and Lower bound. Estimate Trapezoidal estimate independently. Then vary Upper bound alone and observe the new Trapezoidal Rule output. This isolates the changed part of Trapezoidal Rule.
Validating Trapezoidal Rule
Vary Function f(x) alone to test Trapezoidal Rule sensitivity. Freeze Lower bound and watch Trapezoidal estimate. Several simultaneous edits would hide which part of the Trapezoidal Rule setup caused the change in Trapezoidal estimate.
A final Trapezoidal Rule check
Before reusing Trapezoidal estimate, confirm that its label still matches the quantity requested by the original Trapezoidal Rule problem.
Questions about Trapezoidal Rule
Must n be even?
No.
Is this exact for lines?
Yes.
What determines error?
Curvature and panel width.