Area Under Curve Calculator Formula
Understand the math behind the area under curve calculator. Each variable explained with a worked example.
Formulas Used
Area
area = n != -1 ? abs((coeff / (n + 1)) * (pow(x2, n + 1) - pow(x1, n + 1))) : 0Y At X1
y_at_x1 = coeff * pow(x1, n)Y At X2
y_at_x2 = coeff * pow(x2, n)Avg Value
avg_value = x2 != x1 && n != -1 ? abs((coeff / (n + 1)) * (pow(x2, n + 1) - pow(x1, n + 1))) / abs(x2 - x1) : 0Variables
| Variable | Description | Default |
|---|---|---|
coeff | Coefficient (a) | 1 |
n | Exponent (n) | 2 |
x1 | Left Bound (x₁) | 0 |
x2 | Right Bound (x₂) | 4 |
How It Works
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
Worked Example
Find the area under y = x² from x = 0 to x = 4.
coeff = 1n = 2x1 = 0x2 = 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
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