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Pure · Numerical methods

Chapter 1 · 4

The idea

The trapezium rule

Estimating a definite integral by slicing the area into trapezium-topped strips: the formula (h/2)[y₀ + yₙ + 2(y₁ + … + y_(n−1))], choosing h, and deciding whether the estimate is an over- or under-estimate from the curve's shape.

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Pure · Numerical methods

The trapezium rule

Estimating a definite integral by slicing the area into trapezium-topped strips: the formula (h/2)[y₀ + yₙ + 2(y₁ + … + y_(n−1))], choosing h, and deciding whether the estimate is an over- or under-estimate from the curve's shape.

Why it works

When you can't integrate, slice

A definite integral ∫aby dx\int_a^b y\,dx is the area under a curve, and ideally you find it exactly by integrating. But many functions — x2+1\sqrt{x^2 + 1}, 2x2^x, 1+x3\sqrt{1 + x^3} — have no elementary integral. The trapezium rule estimates the area instead, by replacing the awkward curved region with strips you can measure.

The formula: ends once, middles twice

Cut [a,b][a, b] into nn equal strips, each of width h=b−anh = \frac{b - a}{n}. Over one strip the curve is nearly straight, so approximate it by the chord — turning each strip into a trapezium. Adding all nn trapezia:

∫aby dx≈h2[y0+yn+2(y1+⋯+yn−1)]\int_a^b y\,dx \approx \frac{h}{2}\Big[y_0 + y_n + 2(y_1 + \cdots + y_{n-1})\Big]

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