Area Between Two Functions Calculator
The area trapped between two curves is one of the definite integral’s most useful jobs, from consumer surplus in economics to the overlap of two growth projections. The Area Between Two Functions Calculator evaluates it numerically: type both functions, set the interval, and it integrates the absolute difference with Simpson’s rule.
No calculus background is needed to use it. Enter f(x) and g(x) using x, numbers, and functions like sin, cos, exp, log, and sqrt, pick your bounds, and the calculator returns the total enclosed area plus the signed net area, so you can see both the size of the region and which curve sits on top overall.
Quick Answer
Key takeaways
- Total area integrates the absolute difference, so it is never negative even when the curves cross.
- Signed net area integrates f(x) - g(x) and shows which curve dominates overall.
- Type functions naturally: 2x, x^2, sin(x), sqrt(x), and pi all work.
- Numerical integration handles functions with no elementary antiderivative.
How the calculation works
The exact area between two curves is the definite integral of their vertical gap. Because the gap is always measured top-minus-bottom, the absolute value keeps every sub-region positive:
Where curves cross inside [a, b], the absolute value automatically splits the region at each crossing, no manual piecewise setup needed. For the inverse-trigonometry side of function work, see our inverse sine calculator.
Worked example, checked against this exact tool
Enter the pre-filled defaults and click Calculate. The classic textbook region between y = x and y = x2:
| Item | Value |
|---|---|
| f(x) | x |
| g(x) | x^2 |
| Interval | [0, 1] |
| Exact area | ∫(x - x2)dx = 1/2 - 1/3 = 1/6 ≈ 0.166667 |
| Calculator result | 0.166667 square units |
| Signed net area | +0.166667 (f above g throughout) |
Try a crossing pair next: f(x) = sin(x), g(x) = 0 on [0, 2*pi]. The signed net area is 0 (lobes cancel) while the total area is 4, a clean demonstration of why the two numbers differ.
Supported syntax
What competing calculators miss
Most area-between-curves tools accept only polynomials or a fixed function menu. This one is built for real input:
Free-form expressions. Competitors restrict you to preset functions or coefficients. Ours parses full expressions with trig, exponentials, logs, roots, and implicit multiplication.
Both area numbers. Tools typically report one value without saying whether crossings were handled. Ours reports total area and signed net area side by side.
Honest method disclosure. Few tools say how they integrate. Ours names Simpson's rule and the interval count, so the accuracy claim is checkable.
Frequently asked questions
No. Area is a non-negative quantity; the absolute value in the integral guarantees it. If a hand calculation gives a negative number, the curves were misidentified as upper and lower, which is exactly the mistake the absolute value prevents.
The absolute value handles it automatically: each sub-region contributes its positive area. The signed net area will be smaller than the total area whenever crossings occur, since positive and negative lobes partially cancel.
With 2000 subintervals, smooth functions typically match the exact integral to about 6 decimal places. Functions with kinks (like abs) or sharp spikes converge more slowly; narrow the interval around the trouble spot if you need extra digits.
The variable x, numbers, + - * / ^, parentheses, implicit multiplication (2x), functions sin cos tan asin acos atan exp log ln sqrt abs, and the constants pi and e. Powers use ^, so x squared is x^2.
Total area integrates |f - g| and measures the enclosed region's size. Signed area integrates f - g and can cancel; it tells you which function is on top on balance. For sin(x) versus 0 on [0, 2pi], total area is 4 while signed area is 0.
Not for the numeric result; the absolute value accounts for crossings internally. You only need intersection points if you want to set up the integral by hand or split the region for a report.
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References
- Paul Dawkins (Lamar University), "Paul's Online Notes: Area Between Curves" - the integral setup, top-minus-bottom construction, and worked examples. Paul's Online Notes: Area Between Curves. Accessed October 19, 2026.
- OpenStax, "Calculus Volume 1, Section 6.1: Areas Between Curves" - the area theorem for continuous f ≥ g and applied examples. OpenStax: Areas Between Curves. Accessed October 19, 2026.
- Portland Community College (APEX Calculus), "Area Between Curves" - Theorem 7.1.5 with total-area examples including crossing curves. APEX Calculus: Area Between Curves. Accessed October 19, 2026.
- Mathematics LibreTexts, "1.1: Areas Between Curves" - Riemann-sum derivation of the area integral. LibreTexts: Areas Between Curves. Accessed October 19, 2026.
- Khan Academy, "Applications of Integration" (AP Calculus AB) - video and practice on areas between curves. Khan Academy: Applications of Integration. Accessed October 19, 2026.