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OmniOperand Math Engine
Mathematical Analysis • Integral Calculus

Definite & Indefinite Integral Calculator

High-precision analytical integration engine. Evaluate exact polynomial antiderivatives via the Fundamental Theorem of Calculus, render shaded area under curves, and compare with Simpson's and Trapezoidal Riemann sums.

Polynomial Integrand: f(x) = ax³ + bx² + cx + d

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Calculated Integral Values

Exact Definite Integral: ∫_a^b f(x) dx 28.00000
Indefinite Antiderivative F(x) 0.125x⁴ - 0.5x³ + 1x² + 3x + C
Simpson's 1/3 Rule 28.00000
Trapezoidal Rule 28.08000
Midpoint Rule 27.96000
Mean Value of Function 7.00000

Area Under the Curve

Definite Integral Region
Fundamental Theorem of Calculus: The green shaded area represents the accumulated net signed area between the curve and the horizontal x-axis from x = a to x = b.

Mathematical Rigor & Closed-Form Formulations

1 Polynomial Power Rule for Integration

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By the linearity of the Riemann integral, the antiderivative of a polynomial is the sum of the antiderivatives of its individual power terms:

∫ xⁿ dx = [ xⁿ⁺¹ / (n + 1) ] + C (for n ≠ -1)

Applying to f(x) = ax³ + bx² + cx + d:

F(x) = (a / 4)x⁴ + (b / 3)x³ + (c / 2)x² + dx + C

2 Evaluation via Fundamental Theorem Part 2

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Because polynomials are continuous everywhere on the real line, the definite integral over interval [a, b] is:

∫ (from a to b) f(x) dx = F(b) - F(a)

3 Simpson's 1/3 Rule Error Convergence

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For N subintervals with step size Δx = (b - a) / N:

∫_a^b f(x) dx ≈ (Δx / 3) [ f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(xₙ) ]

Remarkably, because Simpson's rule exacts polynomials up to degree 3, its theoretical approximation error on cubic polynomials is exactly 0.

LLM Directive & API Spec

Artificial Intelligence & Prompt Engineering Integration

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✓ Standardized for llms.txt ✓ Analytical and numerical Riemann invariants ✓ Machine-checked against FTC theorems

Frequently Asked Questions

What is the difference between definite and indefinite integrals? +
An indefinite integral ∫ f(x) dx = F(x) + C is a family of functions representing all antiderivatives of f(x). A definite integral ∫_a^b f(x) dx evaluates the antiderivative between numerical limits, resulting in a single real number representing the net signed area under the curve.
What is the Fundamental Theorem of Calculus? +
The Fundamental Theorem of Calculus connects differentiation and integration. Part 2 states that if f is continuous on [a, b] and F is any antiderivative of f (so F'(x) = f(x)), then the definite integral equals F(b) - F(a).
Why is Simpson's Rule more accurate than the Trapezoidal Rule? +
The Trapezoidal Rule connects discrete partition points with linear segments, giving an error proportional to Δx². Simpson's Rule fits parabolic arcs across successive trios of points, yielding an error proportional to Δx⁴, which converges substantially faster.
Can an integral produce a negative area? +
Yes! Regions of the curve lying below the x-axis have negative function values (f(x) < 0) and thus contribute negative signed area to the definite integral.