Area Between Curves Calculator | Two Functions
Calculate the exact area enclosed between two functions f(x) and g(x) over any interval. Automatically detects intersection points, splits the integral correctly, and shows a sub-region breakdown.
Try an example
What Is the Area Between Curves Calculator | Two Functions?
The area between curves is a core topic in integral calculus. It measures the total enclosed space between two functions over a given interval. This calculator correctly handles the most common complications: curves that cross each other within the interval, and area under a single curve (treating the x-axis as the second curve).
- ▸Two-function input: Enter f(x) and g(x) separately. Leave g(x) blank to compute the area under f(x) relative to the x-axis.
- ▸Automatic intersection detection: The calculator scans the interval for sign changes in f(x)−g(x) using bisection, splits the integral at each crossing point, and sums the sub-areas correctly.
- ▸Sub-region breakdown: When curves intersect, a table shows the area of each separate enclosed region and which function is on top in each part.
- ▸Simpson's rule: 1000 subintervals per region give 6–8 significant figures of accuracy for smooth functions.
- ▸Example presets: Four one-click examples (parabola vs line, sin vs cos, x² vs x³, and area under a parabola) to get started instantly.
Formula
A = ∫ₐᵇ |f(x) − g(x)| dx
The absolute value ensures area is always positive, regardless of which function is on top in each sub-region.
A = ∫ₐᵇ [f(x) − g(x)] dx
If the top function never dips below the bottom function in [a,b], no absolute value is needed.
Split at x-values where f(x) = g(x) A = Σ ∫ |f−g| on each sub-interval
When curves cross, f and g swap roles. The integral must be split at each intersection point.
A = ∫ₐᵇ |f(x)| dx (g = 0)
Setting g(x) = 0 (leave blank) computes the area between f(x) and the x-axis.
How to Use
- 1
Enter f(x) | the first function
Type any expression in x: x^2, sin(x), sqrt(x), exp(x), log(x), etc. This is usually (but not always) the upper curve.
- 2
Enter g(x) | the second function (optional)
Type the second function. If you leave it blank, g(x) = 0 and the calculator finds the area between f(x) and the x-axis.
- 3
Set the integration bounds
Enter the lower bound a and upper bound b. These are the x-values that define the left and right edges of the region. Use decimal approximations for irrational values.
- 4
Try a preset example
Click any preset button, "Parabola vs Line", "sin vs cos", etc., to populate the fields with a worked example and see the output format.
- 5
Click Calculate Area and read the result
The total area appears at the top. If the curves intersect, a sub-region table shows each enclosed piece. The step-by-step panel explains how the intersections were found and how the integral was split.
Example Calculation
Example 1 | Parabola vs line: y = x² and y = x, from 0 to 1
Classic result: the area between y=x and y=x² from 0 to 1 is exactly 1/6.
Example 2 | Sine vs cosine: y = sin(x) and y = cos(x), from 0 to π
Example 3 | Area under a parabola: y = x(2−x) from 0 to 2
A classic first-year calculus result, the area under an arch-shaped parabola.
Understanding Area Between Curves | Two Functions
What Is the Area Between Curves?
The area between curves is the total region enclosed between two functions on a graph. It is computed by integrating the difference |f(x) − g(x)| over a specified x-interval. This type of calculation appears throughout calculus courses and has practical applications ranging from physics to economics. The challenge is handling the sign correctly, particularly when curves intersect within the interval.
Why the Absolute Value Matters
A common mistake is to integrate f(x) − g(x) directly without considering which function is on top. If the curves cross, one part of the integral will be positive and another negative, and they will partially cancel, giving the wrong geometric area. The correct approach is to take the absolute value or split at intersection points:
Finding Intersection Points
To find where f(x) = g(x), set f(x) − g(x) = 0 and solve. For polynomial functions, this is often doable algebraically. For transcendental functions, numerical root-finding is needed. This calculator uses a bisection-based scan: it divides the interval into 400 equal segments and detects sign changes in f(x)−g(x). Each sign change is refined by bisection to 60 iterations.
Common Area Between Curves Problems
| f(x) | g(x) | Interval | Exact Area |
|---|---|---|---|
| x | x² | [0, 1] | 1/6 ≈ 0.16667 |
| x² | x³ | [0, 1] | 1/12 ≈ 0.08333 |
| sin(x) | 0 | [0, π] | 2 |
| 4 − x² | x + 2 | [−2, 1] | 4.5 |
| √x | x² | [0, 1] | 1/3 ≈ 0.33333 |
Applications of Area Between Curves
- ▸Physics: Work done by a variable force equals the area under the force-displacement curve. The net work difference between two scenarios is the area between their curves.
- ▸Economics, Consumer and producer surplus: The area between the demand curve and the price line is consumer surplus; the area between the supply curve and price line is producer surplus. Combined, they represent total market welfare.
- ▸Probability: The probability that a random variable falls between two values is the area between the PDF curve and the x-axis over that interval.
- ▸Engineering: Bending moment diagrams, shear force diagrams, and stress-strain curves all involve computing areas between curves.
- ▸Biology: Pharmacokinetic studies use the area under drug concentration curves to compare how different formulations are absorbed and eliminated.
Tips for Accurate Results
- ▸Always check whether the curves cross inside your interval, enter a wide range first if unsure.
- ▸For periodic functions like sin and cos, pick bounds that capture the full enclosed regions (e.g. use intersection points as bounds).
- ▸If you get a surprisingly small area, the curves may be partially cancelling, enable the sub-region table to check.
- ▸For very oscillatory functions, increase accuracy by using a smaller interval and running multiple separate calculations.
Frequently Asked Questions
Why do I need to take the absolute value?
The definite integral ∫[f(x)−g(x)]dx gives a signed result, negative when g(x) > f(x). Geometric area is always positive. Taking the absolute value, or splitting at crossing points where the curves swap roles, ensures you sum positive areas from all sub-regions.
What happens when the curves cross inside the interval?
The calculator automatically detects crossing points by scanning for sign changes in f(x)−g(x) using a bisection algorithm. The interval is split at each crossing, and each sub-region is integrated separately. The results are shown in the sub-region breakdown table.
Can I compute area under a single curve?
Yes, leave g(x) blank (or type 0). The calculator uses g(x) = 0, which means it computes the area between f(x) and the x-axis. Note that if f(x) dips below the x-axis, those parts are still counted positively because of the absolute value.
What function syntax does the calculator accept?
You can use: x^2 (power), sqrt(x), sin(x), cos(x), tan(x), log(x) or ln(x) (natural log), log10(x), exp(x), abs(x), and arithmetic operators +, -, *, /. Constants pi and e are recognised. Example: sin(x)*cos(x), (x^2+1)/(x-2).
How accurate is the result?
Simpson's rule with 1000 subintervals per region gives approximately 6–8 significant figures of accuracy for smooth functions. The intersection-finding bisection uses 60 iterations, giving an accuracy of about 10⁻¹⁸ in the crossing location. For functions with discontinuities or very rapid oscillations, use more caution.
What if I do not know the bounds in advance?
If the curves enclose a region naturally (they intersect on both sides), find the x-coordinates of the intersection points first by setting f(x) = g(x) and solving. You can also use this calculator to help, set a very wide interval and read the intersection points from the step-by-step output, then re-run with those as your bounds.
What is the difference between area between curves and signed area?
Signed area (plain definite integral) can be positive or negative and is useful in physics and economics. Geometric area (area between curves) is always positive and is what this calculator computes. If you specifically need the signed integral ∫f(x)dx (with sign), use an integral calculator instead.