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.
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 absolute value ensures a positive area regardless of which curve lies above the other. This approach applies when each curve is a single power-function term. For more complex functions, the interval should be chosen so the curves do not cross, or the region should be split at intersection points. 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 handles a single interval, so identify sub-intervals first and run it once per section.
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 the result converts the signed integral into a true geometric area. 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.