What is the area between two curves?
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. See also our Laplace Transform Calculator.
What formula is used to calculate the area between curves?
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.
How do I find the integration limits if they are not given?
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.
What happens if the two curves intersect within the interval?
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. You might also find our Summation Calculator (Sigma) useful.
Can this calculator handle sine, cosine, or other trigonometric functions?
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)).
What is the difference between net signed area and total area?
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.
Why does increasing the number of subintervals improve accuracy?
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.
How is the area between curves different from the area under a single curve?
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.