Calculate the area between two curves by entering your upper function f(x), lower function g(x), and the integration bounds (lower limit a and upper limit b). The calculator evaluates the definite integral of f(x) − g(x) over your specified interval and returns the enclosed area along with a visual breakdown of the region.
Area Between Curves
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Integral of f(x) — Upper Curve Area
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Integral of g(x) — Lower Curve Area
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Intersections Found in Interval
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The area between two curves f(x) and g(x) over an interval [a, b] is the definite integral of the absolute difference |f(x) − g(x)| from a to b. It represents the total area of the region sandwiched between the two functions on that interval.
The standard formula is A = ∫[a to b] (f(x) − g(x)) dx, where f(x) is the upper curve and g(x) is the lower curve. If the curves cross within the interval, you must split the integral at the intersection points and sum the absolute values of each sub-region.
When limits are not specified, you first find where the two curves intersect by setting f(x) = g(x) and solving for x. The x-values of those intersection points become your lower and upper limits of integration.
If f(x) and g(x) cross inside [a, b], one curve becomes dominant in each sub-interval. You must break the integral at each crossing point and integrate |f(x) − g(x)| on each sub-interval separately, then add the results. This calculator detects crossings and computes the total absolute area.
Yes. Select the 'Sine vs x-axis' preset to compute the area under a sine curve. For fully custom trigonometric functions, adjust the limits to match the relevant period of the function (e.g., 0 to π for one arch of sin(x)).
Net signed area allows positive and negative contributions to cancel each other out, while total area sums the absolute value of each portion. This calculator returns the total (absolute) area between the curves, which is always non-negative.
This calculator uses numerical integration (Simpson's rule approximation). More subintervals means smaller steps along the x-axis, reducing the approximation error. For smooth functions, 1000 subintervals typically gives results accurate to 5–6 decimal places.
Area under a single curve is ∫f(x)dx measured from the x-axis (g(x) = 0). Area between two curves is ∫(f(x) − g(x))dx, effectively subtracting the region under the lower curve from the region under the upper curve to isolate the enclosed space between them.