Area Under Curve Calculator

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Technical Review by: David Chen, CFA, Data Science Specialist
Last Updated: October 2023 | Verified for Mathematical Accuracy

Determine the definite integral of a dataset using our Area Under Curve Calculator. Whether you are analyzing medical ROC curves, financial trends, or physical motion, this tool provides precise results using the reliable Trapezoidal Rule.

Area Under Curve Calculator

Enter one pair per line, separated by a comma. X values should be in increasing order.

Total Area (AUC)

0.00
Calculated via Trapezoidal Rule

Area Under Curve Calculator Formula

Area = Σ [ (yᵢ + yᵢ₊₁) / 2 ] × (xᵢ₊₁ - xᵢ) Formula Source: Wolfram MathWorld

Variables:

  • xᵢ: The independent variable (e.g., time, concentration, or index).
  • yᵢ: The dependent variable (the value of the function at xᵢ).
  • (xᵢ₊₁ – xᵢ): The width of the sub-interval.
  • (yᵢ + yᵢ₊₁) / 2: The average height of the trapezoid.

Related Calculators:

What is Area Under Curve Calculator?

An Area Under Curve (AUC) calculator is a mathematical tool used to estimate the total “accumulation” or “integral” of a set of data points. In statistics, it is widely used for ROC curves to measure a model’s diagnostic power. In physics, the area under a velocity-time graph represents total displacement.

The Trapezoidal Rule is the most common numerical method for this. It works by approximating the region under the graph as a series of trapezoids rather than rectangles, providing much higher accuracy for curves that are not perfectly flat.

How to Calculate Area Under Curve (Example):

  1. List your data points as (X, Y) coordinates.
  2. Ensure X values are sorted from smallest to largest.
  3. For each pair of adjacent points, find the average height: (Y1 + Y2) / 2.
  4. Multiply the average height by the width: (X2 – X1).
  5. Sum all individual trapezoid areas to get the total AUC.

Frequently Asked Questions (FAQ):

Why is AUC used in medicine?

In pharmacology, AUC represents the total exposure of a drug in the bloodstream over time, helping doctors determine the correct dosage.

Can the area be negative?

If the curve falls below the X-axis (negative Y values), the mathematical integral will be negative, representing “net area.”

Is the Trapezoidal Rule accurate?

It is significantly more accurate than left-point or right-point Riemann sums. Accuracy increases as you add more data points.

Does order of points matter?

Yes, X-coordinates must be strictly increasing for the standard AUC calculation to represent a physical area correctly.

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