Area Under Curve Calculator
Calculate the area between a curve and the x-axis over any interval using Simpson's rule numerical integration.
Area Under Curve Calculator
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Formula: ∫[a,b] f(x) dx ≈ Σ f(xᵢ)Δx (Simpson's Rule)
Worked example — Area = 9.000000
Formula
∫[a,b] f(x) dx ≈ Σ f(xᵢ)Δx (Simpson's Rule)
The area between a curve and the x-axis over [a, b] is approximated by dividing the interval into many small subintervals, fitting a parabola through each group of three points, and summing the resulting parabolic-segment areas — Simpson's rule.
Worked Examples
Example 1: Area under a parabola
Problem:Find the area under f(x) = x² between x = 0 and x = 3.
Solution:∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9.
Result:Area = 9.000000
Example 2: Distance from a velocity curve
Problem:A car's velocity (m/s) over time (s) is modeled by v(t) = 2t + 1. Find the distance traveled between t = 0 and t = 4 seconds.
Solution:Distance = ∫₀⁴ (2t + 1) dt = [t² + t]₀⁴ = (16 + 4) − 0 = 20.
Result:Distance = 20 meters
Frequently Asked Questions
What does 'area under the curve' actually represent?
The area under a curve between two x-values is the definite integral of the function over that interval — geometrically, it's the signed area enclosed between the curve, the x-axis, and the two vertical boundary lines. Area above the x-axis counts as positive; area below counts as negative, so a function that dips below zero can partially cancel out area gained elsewhere in the interval.
How does Simpson's rule approximate the area more accurately than simply averaging heights?
Instead of assuming the curve is flat (rectangles) or straight (trapezoids) between sample points, Simpson's rule fits a parabola through every three consecutive points and finds the exact area under that parabola. Because most smooth functions locally resemble a parabola quite closely, this gives dramatically better accuracy — error shrinks proportional to the fourth power of the interval width, versus the second power for the trapezoidal rule.
What is 'area under the curve' (AUC) used for in medicine and diagnostics?
In diagnostic testing, the ROC (Receiver Operating Characteristic) curve plots true positive rate against false positive rate, and its AUC score (0.5 = no better than chance, 1.0 = perfect) summarizes how well a test distinguishes patients with a condition from those without it. In pharmacology, AUC of a drug-concentration-vs-time curve estimates total drug exposure in the bloodstream, guiding dosing decisions.
How is area under the curve applied in economics and finance?
Consumer surplus and producer surplus in economics are literally areas under (or above) supply and demand curves, representing the total benefit to buyers or sellers. In finance, the area under a probability density function over a range of returns gives the probability of a portfolio falling within that return range.
What is the connection between area under a velocity curve and distance traveled?
If you plot an object's velocity against time, the area under that curve between two times equals the total displacement over that interval — this is a direct physical application of the Fundamental Theorem of Calculus, since velocity is the derivative of position, and integrating it back recovers the distance traveled.
Why might my area calculation be inaccurate for certain functions?
Functions with discontinuities, vertical asymptotes, or extremely rapid oscillation within the integration interval can defeat numerical methods like Simpson's rule, which assume the function is reasonably smooth between sample points. If your function has a singularity (like 1/x crossing zero) inside [a, b], the result may be wildly inaccurate or meaningless.
How many sample points does Area Under Curve Calculator use, and does adding more always help?
Area Under Curve Calculator uses 1,000 subintervals by default, giving excellent precision (often 6+ correct decimal digits) for smooth functions. Beyond a certain point, adding more subintervals yields diminishing returns and can even introduce floating-point rounding error, so 1,000 is a practical balance between speed and accuracy for most everyday functions.
How do you get the total geometric area when the curve crosses the x-axis?
The definite integral returns signed area, so a section below the axis subtracts from one above and the two can cancel. To get the total unsigned area, find the roots of f inside the interval, integrate between consecutive roots, take the absolute value of each piece, and add them. For sin(x) from 0 to 2 pi the signed integral is 0 while the total area is 4, since the two humps have area 2 each and opposite signs.
What is the 1, 4, 2, 4, 1 weighting pattern in Simpson's rule?
Composite Simpson's rule multiplies the sampled heights by weights of 1 at the two endpoints, 4 at every odd-indexed interior point and 2 at every even-indexed one, then scales the sum by the step width divided by 3. The pattern comes from stitching together parabolas across successive pairs of subintervals, which is why the number of subintervals must be even. The rule is exact for any polynomial up to cubic, which is where its fourth-order accuracy comes from.
What happens if the bounds are reversed or one of them is infinite?
Reversing the limits flips the sign, since the integral from b to a equals minus the integral from a to b; this tool requires the lower bound to be strictly less than the upper bound, so negate the result yourself if you need the other orientation. An infinite bound makes the integral improper and no fixed-step rule can handle it directly. Rewrite it as a limit, or substitute a variable such as x equal to 1/t to map the infinite tail onto a finite interval before integrating.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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