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Area by Definite Integral Online Calculator

Area-by-integration calculator. Enter f(x) and an interval to get both the net integral and the geometric area (absolute value), separating signs—for integral geometry.

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A = ∫ₐᵇ |f(x)| dx
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Area by Definite Integral

1. Definition
The geometric area between y=f(x) and the x-axis over [a,b] is A=∫ₐᵇ |f(x)|dx, while ∫ₐᵇ f(x)dx is the signed net integral.
2. Symbols
SymbolsMeaning
∫f dxNet (signed) integral
∫|f| dxGeometric area (always ≥0)
Below x-axisNegative for net, positive for area
3. How it works
  • Numerically computes both the net integral and the absolute area;
  • The difference shows how much lies below the x-axis;
  • It distinguishes “integral value” from “actual area”.
4. Steps
  1. Enter the curve f(x);
  2. Enter the interval [a,b];
  3. Click Calculate for the net and geometric area.
5. Example
Example: f(x)=sin x on [0,π]. Both ∫sin x and ∫|sin x| equal 2.
6. Pitfalls
Where the curve crosses the axis, the net integral cancels;
True area needs |f|;
Area between two curves uses ∫|top−bottom|.

❓ FAQ | Area by Definite Integral

Why do net integral and area differ?
Below the x-axis the net integral subtracts, while area counts it positively.
Area between two curves?
A=∫|top−bottom|dx.
Can area be negative?
Geometric area is ≥0; the net integral may be negative.
What integrator is used?
Adaptive Simpson on both f and |f|.
Which functions?
+−*/^ and common elementary functions.