Trapezoidal Rule with sample values
For x² on 0 to 3, increasing n drives the estimate toward 9.
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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
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Why this relationship works
The trapezoidal rule averages neighboring endpoint heights and multiplies by panel width.
For this page, trapezoidal rule is interpreted under the stated convention and input order.
Manual method
Weight endpoints by one half, interior samples fully, and multiply their sum by h. Trapezoidal Rule also leads to Simpson comparison.
Where the shortcut stops
Strong curvature or discontinuities require more panels; straight chords bias convex and concave regions differently.
Compare the result with the worked example's scale before relying on the reported trapezoidal estimate.
Where this calculation appears
It integrates tabulated data and smooth functions without derivative formulas. A related application of Trapezoidal Rule is rectangle sums.
A closer look at result behavior
The cleanest input audit is to restate the problem using Function f(x), Lower bound, Upper bound and Subinterval count. If that sentence sounds wrong, correct the assignment before asking for trapezoidal estimate.
Doubling n halves panel width and usually reduces smooth-function error substantially. The pattern also provides a quick estimate of whether a revised result is plausible.
The formula and result serve different readers: the formula shows what was done, and the rounded trapezoidal estimate communicates scale. Keep both when the work needs review.
Simpson’s rule uses quadratic panels and requires even n. The wording of the problem should decide which operation is appropriate.
A screenshot is unnecessary when the shown formula and the Trapezoidal Rule entries (Function f(x), Lower bound, Upper bound and Subinterval count) are saved in plain text. Include the rounding rule if the reported trapezoidal estimate is approximate.
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.
Questions about Trapezoidal Rule
Must n be even?
No.
Is this exact for lines?
Yes.
What determines error?
Curvature and panel width.