Area Under Curve Calculator
Calculate the area under y = ax^n between two x-values using the definite integral.
Fläche
21.3333
Formel
## Area Under a Curve ### Definite Integral The area under y = ax^n from x₁ to x₂ is: **Area = |integral from x₁ to x₂ of ax^n dx|** **= |(a/(n+1)) × [x₂^(n+1) - x₁^(n+1)]|** ### Average Value The average value of f on [x₁, x₂] is: **f_avg = (1/(x₂ - x₁)) × integral of f(x) dx**
Lösungsbeispiel
Find the area under y = x² from x = 0 to x = 4.
- 01Antiderivative: x³/3
- 02F(4) = 64/3 ≈ 21.3333
- 03F(0) = 0
- 04Area = 21.3333 - 0 = 21.3333
- 05Average value = 21.3333 / 4 ≈ 5.3333
Häufig Gestellte Fragen
What does the area under a curve represent?
Geometrically, it is the area between the function and the x-axis. In applications, it can represent distance (under velocity), work (under force), or accumulated quantity.
What about area below the x-axis?
The definite integral gives a signed area: positive above the axis, negative below. This calculator shows the absolute area.
How is this related to the integral?
The definite integral from a to b of f(x) dx gives the net signed area. The Fundamental Theorem of Calculus connects this to the antiderivative.
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