How Definite Integrals Are Computed
A definite integral ∫ab f(x) dx measures the signed area under the curve between x = a and x = b. The Fundamental Theorem of Calculus says: find an antiderivative F, then compute F(b) − F(a). When no elementary antiderivative exists, Simpson's rule approximates the area by fitting parabolas through sample points — with 1000 subintervals, the result is accurate far beyond what homework requires.
Simpson's Rule in One Line
Split [a, b] into an even number n of slices of width h. Then ∫f ≈ (h/3)·[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(xₙ)]. The alternating 4-2-4 weights are what make it far more accurate than simple rectangles or trapezoids.
Exact vs. Numeric
Where the antiderivative is a standard form (powers, sin, cos, e^x), this tool also shows the exact value for comparison. For symbolic antiderivatives in general, try our indefinite integral calculator.