Midpoint rectangles, trapezoid roofs, and Simpson arcs over the same partitionLive current inputs
Math & Statistics
Integral Value Table Calculator
Generate an interval-by-interval value table for a quadratic integrand and reconcile left, right, midpoint, trapezoid, and Simpson composite rules against its exact antiderivative.
Exact integral -
Composite Simpson result -
Composite trapezoid result -
Composite midpoint result -
Composite left result -
Composite right result -
Simpson absolute error -
Subinterval width -
QUADRATURE STRIP ATLAS
Midpoint rectangles, trapezoid roofs, and Simpson arcs over the same partition
The diagram draws a representative set of live slices and uses a cumulative error rail to show which rule is over or under the exact quadratic area.
INTERVAL-BY-INTERVAL LEDGER
Nodes, midpoints, function values, and running trapezoid area
Every row corresponds to one physical subinterval rather than repeating only the final totals.
| Interval | Bounds | f(left) | f(mid) | f(right) | Trapezoid slice | Running trapezoid |
|---|
TABLE SETUP
Hold the partition fixed while comparing integration rules
- Enter the quadratic coefficients.
- Set the integration bounds.
- Choose an even n so Simpson's rule is defined.
- Inspect slice-level values before comparing final errors.
- Increase n to study convergence rather than to conceal a wrong integrand.
CURVATURE AND BIAS
Convexity determines which simple rules overestimate or underestimate
For a convex quadratic, trapezoids tend to sit above the curve while midpoint rectangles tend to sit below it. Reversing curvature reverses those biases.
Simpson's parabolic interpolation matches a quadratic exactly, making this page a transparent validation case for an implementation.
COMPOSITE QUADRATURE
Use one partition to compare rules without changing the evidence
All rules use the same bounds and n equal subintervals. Simpson combines endpoint and midpoint information in pairs and is exact for the entered quadratic apart from floating-point rounding.
Detailed calculation process and general formulas
dx=(u-l)/nL=dx sum_(i=0)^(n-1) f(x_i)M=dx sum_(i=0)^(n-1) f(x_i+dx/2)T=dx[f(x_0)/2+sum f(x_i)+f(x_n)/2]S=(dx/3)[f_0+f_n+4 sum f_odd+2 sum f_even]Symbols, meanings, and units
- n
- even number of equal subintervalscount
- dx
- subinterval widthx-unit
- L,R
- left and right rectangle sumsf-unit*x-unit
- M
- midpoint composite sumf-unit*x-unit
- T
- composite trapezoid resultf-unit*x-unit
- S
- composite Simpson resultf-unit*x-unit
RULE COMPARISON
The table explains why the totals differ
Global error is built from local slice geometry.
Partition scale
-dx shows the resolution shared by all methods.
Curvature-sensitive pair
-Midpoint and trapezoid usually bracket a smooth quadratic integral.
Polynomial exactness
-Simpson should reconcile to the analytic quadratic integral.
Decision takeaway: Use slice rows to diagnose a rule; a final total alone cannot reveal indexing or endpoint-weight mistakes.
IMPLEMENTATION AUDIT
Checks for spreadsheet or code quadrature
- n is even for Simpson's rule.
- There are exactly n+1 endpoint nodes.
- Interior Simpson weights alternate 4 and 2.
- The right endpoint is included once.
- All rules share the same dx and function values.
Applied decisions
Quadrature comparisons that benefit from a live ledger
Spreadsheet validation
A composite integration workbook is tested on a quadratic.
What the result clarifies: Simpson exactness and row-level weights expose off-by-one errors.
Sampling plan
A smooth response is measured at an adjustable number of equally spaced stations.
What the result clarifies: The partition view links added samples to smaller local slice width.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
Simpson's exactness here is specific to polynomials of degree three or less and exact arithmetic. It does not guarantee exact results for arbitrary measured functions.
Integral Value Table Calculator | Composite Quadrature Ledger FAQ
Why must n be even?
Composite Simpson's rule applies parabolic panels across pairs of subintervals.
Why include both left and right sums?
They show directional endpoint bias and provide a simple implementation check.
Should midpoint and trapezoid always bracket the exact value?
Not for every function, but they do predictably for a quadratic with consistent curvature.
Why is Simpson error not exactly zero sometimes?
Floating-point arithmetic and formatted output can leave a tiny residual.