Definite Integral (Polynomial)
∫ₐᵇ p(x) dx using power rule for antiderivatives.
Result
∫₍0₎^(2) f(x) dx
4.000000
Area under f(x) from 0 to 2.
- f(x)1x³ + 0x² + 0x + 0
- Antiderivative F(x)(1/4)x⁴ + (0/3)x³ + (0/2)x² + 0x
- F(b)4.000000
- F(a)0.000000
- F(b) − F(a)4.000000
Step-by-step
- Antiderivative via power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1).
- F(x) = (1/4)x⁴ + (0/3)x³ + (0/2)x² + 0x.
- Definite integral = F(2) − F(0) = 4.0000 − 0.0000 = 4.0000.
How to use this calculator
- Enter coefficients up to x³ + bounds a, b.
About this calculator
Definite integral = signed area under curve from a to b. Fundamental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x). Power rule for antiderivative: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. For polynomials, integration is mechanical: increment exponent, divide by new exponent. Used for area, work, displacement from velocity, average value. Source: NIST DLMF, Wolfram MathWorld - Definite Integral.
Frequently asked
Definite: number (area between bounds). Indefinite: function (antiderivative + C).
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