Positive lobes and negative lobes reconciled against one antiderivative jumpLive current inputs
Math & Statistics
Integral Solver Calculator
Solve a definite integral of a cubic polynomial exactly, separate positive and negative geometric area at real zero crossings, and reconcile net area with antiderivative endpoints.
Signed definite integral -
Positive area -
Negative area magnitude -
Total geometric area -
Average function value -
Zero crossings in interval -
Antiderivative at lower bound -
Antiderivative at upper bound -
SIGNED-AREA MAP
Positive lobes and negative lobes reconciled against one antiderivative jump
The graph fills above-axis and below-axis regions with different colors, labels in-window roots, and preserves the distinction between net and geometric area.
ROOT-SPLIT AREA LEDGER
Every sign-stable interval integrated exactly
The solver partitions the requested bounds at detected roots, then evaluates the same analytic antiderivative on every segment.
| Segment | Bounds | Midpoint sign | Signed area | Geometric contribution |
|---|
SOLVER SETUP
Keep signed accumulation separate from physical area
- Enter all four cubic coefficients.
- Enter bounds in either order; the solver orders them for geometric analysis.
- Use the signed result for net accumulation.
- Use positive, negative, and geometric results when cancellation matters.
- Inspect detected roots before interpreting one continuous process as multiple regimes.
WHY ROOTS MATTER
Two large lobes can produce a deceptively small net integral
A negative integrand subtracts from signed accumulation but still occupies geometric area. Reporting both views prevents cancellation from being mistaken for little activity.
The endpoint antiderivative difference is the exact reconciliation. The lobe table is a decomposition of that same result, not a separate approximation.
FUNDAMENTAL THEOREM
Build the antiderivative once, then evaluate and split only for interpretation
The definite integral is F(upper)-F(lower). Root splitting is not needed for net area, but it is required to report positive and negative geometric contributions correctly.
Detailed calculation process and general formulas
f(x)=ax^3+bx^2+cx+dF(x)=ax^4/4+bx^3/3+cx^2/2+dxI=F(u)-F(l)A_geom=sum |F(r_(i+1))-F(r_i)|f_avg=I/(u-l)Symbols, meanings, and units
- l,u
- ordered integration boundsx-unit
- f
- integrandf-unit
- F
- one antiderivative of ff-unit*x-unit
- I
- signed definite integralf-unit*x-unit
- A_geom
- sum of absolute lobe areasf-unit*x-unit
- f_avg
- average value over the intervalf-unit
AREA INTERPRETATION
Choose the result that matches the physical question
Net change, total exposure, and average level are different decisions.
Net accumulation
-Use the signed integral when negative rates reverse positive rates.
Total exposure
-Use geometric area when magnitude matters regardless of direction.
Equivalent steady level
-Average value spreads the net integral evenly across the interval.
Decision takeaway: State whether cancellation is meaningful before choosing signed or geometric area.
MODEL BOUNDARIES
Checks before the cubic area enters a real decision
- Coefficient and x units are documented.
- The interval lies within the fitted model range.
- Negative values have a physical interpretation.
- Root locations are not artifacts of extrapolation.
- Required absolute-value exposure is reported separately.
Applied decisions
One curve, three different area questions
Inventory rate
Positive production and negative depletion occur in one modeled window.
What the result clarifies: Net area gives inventory change while geometric area gives total throughput.
Signed power
A bidirectional system alternates delivery and absorption.
What the result clarifies: The lobe ledger retains both energy directions before netting them.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The exact integral is exact for the entered cubic, not necessarily for the real process that the cubic approximates. Root finding and display values use finite precision.
Integral Solver Calculator | Signed Area of a Cubic Polynomial FAQ
Why can geometric area exceed the signed integral?
Negative lobes subtract from signed area but add by magnitude to geometric area.
What happens if I reverse the bounds?
The displayed signed integral follows the entered orientation while geometric decomposition uses ordered bounds.
Does the antiderivative need +C?
Any constant cancels in F(u)-F(l), so zero is used.
How are roots found?
The interval is scanned and bracketed sign changes are refined numerically, with near-zero points consolidated.