Area Under Curve Calculator
Free Area under curve Calculator for statistics. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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