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Back to Integral Calculus

Numerical Integration

Most integrals have no formula, so computers use rules. Trapezoids, midpoints and parabolas all converge — but at very different speeds.

sin x on [0, π], n = 2 · step 1 of 11

Trapezoid: 1.570796

Midpoint: 2.221441

Simpson: 2.094395

Error against n (log scales): trapezoid, midpoint, Simpson

Speed

∫ sin x dx from 0 to π is exactly 2. A computer cannot take a limit, so it settles for a finite rule. Three classic ones, side by side: trapezoids, midpoint rectangles, and Simpson's parabolas.

T_{2} = \tfrac{\Delta x}{2}\big[f_0 + 2f_1 + \cdots + 2f_{1} + f_{2}\big] = 1.570796

M_{2} = \Delta x\big[f(m_1) + \cdots + f(m_{2})\big] = 2.221441

S_{2} = \tfrac{\Delta x}{3}\big[f_0 + 4f_1 + 2f_2 + \cdots + 4f_{1} + f_{2}\big] = 2.094395

\int_0^{\pi} \sin x\,dx = 2, \qquad \Delta x = 1.5708

Numerical Integration

Most integrals have no formula, so computers use rules. Trapezoids, midpoints and parabolas all converge — but at very different speeds.

sin x on [0, π], n = 2 · step 1 of 11

Trapezoid: 1.570796

Midpoint: 2.221441

Simpson: 2.094395

Error against n (log scales): trapezoid, midpoint, Simpson

Speed

∫ sin x dx from 0 to π is exactly 2. A computer cannot take a limit, so it settles for a finite rule. Three classic ones, side by side: trapezoids, midpoint rectangles, and Simpson's parabolas.

T_{2} = \tfrac{\Delta x}{2}\big[f_0 + 2f_1 + \cdots + 2f_{1} + f_{2}\big] = 1.570796

M_{2} = \Delta x\big[f(m_1) + \cdots + f(m_{2})\big] = 2.221441

S_{2} = \tfrac{\Delta x}{3}\big[f_0 + 4f_1 + 2f_2 + \cdots + 4f_{1} + f_{2}\big] = 2.094395

\int_0^{\pi} \sin x\,dx = 2, \qquad \Delta x = 1.5708

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Numerical Integration

See also

Numerical Integration

See also