Mathematical Analysis • Integral Calculus
Määrätyn Integraalin & Integraalilaskennan Peruslauseen Laskuri
Laske polynomien integraalit ja määrätyt integraalit analyyttisellä tarkkuudella.
Polynomial Integrand: f(x) = ax³ + bx² + cx + d
a
b
c
d
a
b
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
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
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
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.
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Usein Kysytyt Kysymykset
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.