Area Between Curves Calculator
Finds the enclosed area between two power-function curves over a chosen interval. Useful in calculus courses when computing regions bounded by f(x) = ax^n and g(x) = bx^m.
Last updated: September 2026
Formula below · 2 sources (mathworld.wolfram.com, Wikipedia) · Updated Sep 2026
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About this calculator
The area between two curves f(x) = a·xⁿ and g(x) = b·xᵐ over [L, U] is found by integrating their difference. Using the power rule for integration, ∫xⁿ dx = xⁿ⁺¹/(n+1), the formula becomes: Area = |[a/(n+1)]·(U^(n+1) − L^(n+1)) − [b/(m+1)]·(U^(m+1) − L^(m+1))|. The calculator finds every point in [L, U] where the curves cross, integrates each piece separately and adds the absolute values, so the result is the true enclosed area even when the curves swap places (for 2x² and x on [0, 2], which cross at x = 0.5, that is 3.4167 rather than the net 3.3333). Each curve must be a single power-function term. This calculator handles one term per curve, making it ideal for quick checks in introductory calculus.
How to use
Suppose f(x) = 3x² and g(x) = x³, integrated from x = 0 to x = 2. Enter Curve 1 Coefficient = 3, Curve 1 Exponent = 2, Curve 2 Coefficient = 1, Curve 2 Exponent = 3, Lower Limit = 0, Upper Limit = 2. Compute: [3/3]·(8 − 0) = 8 for curve 1; [1/4]·(16 − 0) = 4 for curve 2. Area = |8 − 4| = 4 square units. This confirms the region enclosed between x² and x³ on [0, 2] has area 4.
Frequently asked questions
How do I find the area between two curves that intersect?
When two curves intersect, they swap which one is on top, so you must split the integral at each intersection point. Find the x-values where f(x) = g(x), then compute the area on each sub-interval separately, always subtracting the lower curve from the upper one. Sum the absolute values of each piece to get the total enclosed area. This calculator does that splitting for you automatically.
Why does the area between curves formula use an absolute value?
The definite integral of f(x) − g(x) can be negative if g lies above f on the interval, which would give a geometrically meaningless negative area. Taking the absolute value of each piece between crossing points converts the signed integral into a true geometric area; this calculator does that piece by piece. If you already know which function is larger, you can subtract the smaller from the larger to avoid ambiguity. The absolute value is simply a safeguard that works regardless of curve ordering.
What is the difference between area under a curve and area between two curves?
The area under a single curve measures the region between that curve and the x-axis (g(x) = 0). The area between two curves measures the region sandwiched between f(x) and g(x), effectively treating one curve as the baseline instead of the x-axis. Mathematically, it is ∫[f(x) − g(x)] dx rather than ∫f(x) dx. If g(x) = 0, the two concepts are identical, so area under a curve is a special case of area between curves.