Definite Integral Calculator
Compute the exact or approximate area under a polynomial curve between two bounds. Ideal for calculus students and engineers evaluating net change, area, or accumulated quantities.
About this calculator
The definite integral ∫[a to b] f(x) dx measures the signed area between a curve and the x-axis from x = a to x = b. For a polynomial f(x) = c₁x + c₂x² + … + cₙxⁿ, the antiderivative is F(x) = c₁x²/2 + c₂x³/3 + … + cₙx^(n+1)/(n+1). The Fundamental Theorem of Calculus states the exact result is F(b) − F(a). When an exact symbolic answer is needed, this method is used. Alternatively, numerical methods approximate the integral: the Trapezoidal Rule divides [a, b] into n equal strips and averages adjacent function values, while Simpson's Rule fits parabolic arcs over pairs of strips for higher accuracy. The Trapezoidal approximation is ≈ h × [f(a)/2 + f(x₁) + … + f(xₙ₋₁) + f(b)/2] and Simpson's is ≈ (h/3)[f(a) + 4f(m) + f(b)] for a single interval, where h = (b − a)/n.
How to use
Integrate f(x) = 3x² from x = 1 to x = 4. Enter coefficients '3' (meaning 3x¹, but here treating as 3x²: enter '0,3' for c₁=0, c₂=3), lower bound 1, upper bound 4, method 'exact'. The antiderivative is F(x) = x³. Evaluate: F(4) − F(1) = 64 − 1 = 63. So ∫[1 to 4] 3x² dx = 63. You can verify with the trapezoidal method for comparison — it will yield a close numerical approximation.
Frequently asked questions
What is the difference between exact integration and numerical integration methods?
Exact integration uses the Fundamental Theorem of Calculus: find the antiderivative F(x) and compute F(b) − F(a), giving a precise symbolic answer with no rounding error. Numerical methods like the Trapezoidal Rule and Simpson's Rule approximate the integral by summing geometric shapes under the curve. Numerical methods are useful when an antiderivative is difficult or impossible to express in closed form, but they introduce a small approximation error that decreases as the number of subdivisions increases.
How accurate is Simpson's Rule compared to the Trapezoidal Rule for definite integrals?
Simpson's Rule is generally more accurate than the Trapezoidal Rule for smooth functions because it fits parabolic arcs rather than straight-line segments between points. The error in the Trapezoidal Rule is proportional to h² (where h is the step size), while Simpson's Rule error is proportional to h⁴, making it converge much faster as you increase subdivisions. For polynomials of degree ≤ 3, Simpson's Rule is actually exact. For practical purposes, Simpson's Rule with even a modest number of intervals is highly precise.
How do you enter polynomial coefficients for a definite integral calculator?
Coefficients are entered as a comma-separated list ordered by ascending power. The first value is the coefficient of x¹, the second of x², and so on. For example, to integrate 2x + 5x³, you would enter '2, 0, 5' since the x² coefficient is zero. Constant terms (degree 0) are handled separately if the calculator supports them. Always verify the order before calculating to ensure the correct polynomial is integrated.