Calculus
Integration by approximation
Trapezoids and Simpson's rule — when antiderivatives won't budge.
When an antiderivative isn't elementary, approximate the area numerically.
Try this
4
midpoint-rectangle estimate of ∫₀¹ x² dx (true value 1/3 ≈ 0.333) = 0.33
Why bother approximating?
Some integrals have no elementary antiderivative at all — ∫ e^(−x²) dx (the bell curve), ∫ sin(x)/x dx, ∫ √(1 + x³) dx. The Fundamental Theorem is useless there, so we slice the area into shapes we *can* measure and add them up.
From crude to sharp
- Rectangles (Riemann sum) — left, right, or midpoint heights; midpoint is usually the most accurate of the three.
- Trapezoid rule — joins the tops with straight lines; beats plain rectangles.
- Simpson's rule — fits little parabolas through triples of points; far more accurate for the same number of slices.
- More slices ⇒ closer to the truth — watch the slider home in on 1/3 ≈ 0.333.
Your turn
Estimate ∫₀² x dx with 2 rectangles of width 1 using right endpoints.
Common methods
- Rectangle (Riemann) — sum of f(xᵢ)·Δx.
- Trapezoidal — averages left and right rectangles.
- Simpson's — fits parabolas through triples; very accurate.