Formula used
The signed integral counts regions below the reference curve negatively. Geometric area uses absolute region sizes instead. Splitting at roots prevents cancellation between separate regions.
How to use this calculator
- Select area under one curve, area between curves, or coordinate data.
- Enter valid functions using x as the independent variable.
- Set the lower and upper limits for equation modes.
- Choose a numerical method and suitable subinterval count.
- Select radians or degrees for trigonometric function inputs.
- Choose output precision, units, steps, and graph options.
- Press Calculate area and review warnings before using results.
- Copy, print, or download the result as CSV or PDF.
Worked examples
| Purpose | First curve | Second curve | Interval | Expected result |
|---|---|---|---|---|
| Polynomial area | x^2 | 0 | 0 to 3 | 9 square units |
| Sine area | sin(x) | 0 | 0 to pi | 2 square units |
| Between curves | x | x^2 | 0 to 1 | 1/6 square unit |
| Crossing axis | x^2 - 1 | 0 | -2 to 2 | Signed and geometric areas differ |
Understanding curve area
An area-under-a-curve problem measures accumulated vertical slices across an interval. A definite integral combines those slices while tracking their signs. Regions below the x-axis therefore reduce the signed total.
Geometric area answers a different practical question. It adds each region using a positive magnitude. The calculator detects roots and intersections to separate those regions.
For two curves, the relevant height is their vertical difference. The order of the functions controls the signed result. Geometric area remains positive after each region is separated.
Exact integration is available only for recognized simple polynomial input here. Other expressions use dependable numerical approximation methods. Increasing subdivisions generally improves smooth-function accuracy, but discontinuities still require caution.
Riemann sums use rectangles placed at selected sample points. The trapezoidal rule connects neighboring values with straight lines. Simpson methods fit parabolic behavior and often converge faster.
Adaptive Simpson integration concentrates effort where the curve changes rapidly. It can be efficient for smooth functions with localized curvature. Undefined points may still make the integral improper or divergent.
Coordinate mode is useful for measurements, experiments, and sampled signals. The tool sorts x-values before applying trapezoids. Duplicate x-values are rejected because interval width becomes zero.
Numerical error is estimated by comparing two subdivision levels. This difference is an indicator, not a formal proof. Always inspect the graph and warnings for unexpected behavior.
Frequently asked questions
What is signed area?
Signed area treats regions above the reference as positive and regions below it as negative.
What is geometric area?
Geometric area adds absolute region sizes, so separate positive and negative regions never cancel.
Which numerical method should I choose?
Simpson's 1/3 rule is a strong default for smooth functions. Trapezoidal mode suits tabulated data.
Why did my subinterval count change?
Simpson's 1/3 rule needs an even count. Simpson's 3/8 rule needs a multiple of three.
Can the calculator detect intersections?
Yes. It scans the selected interval, detects sign changes, and refines those locations with bisection.
Can it solve improper integrals?
It may detect undefined samples, but it does not perform formal limit analysis for improper integrals.
Why does a negative base fail with a fractional power?
That operation can produce a complex value, while this calculator is limited to real-valued curves.
How are coordinate points integrated?
Adjacent points form trapezoids. Sign-changing segments are split at a linearly interpolated zero crossing.
Does a smaller error estimate guarantee correctness?
No. It only compares two numerical resolutions and cannot prove convergence near every discontinuity.