Numerical integration and differentiation · Definition

Trapezoidal rule

numerical methodsnumerical integration and differentiation

Before you start you need to know

It is one of the first numerical integration rules studied. Its geometric idea is simple: add up trapezoid areas instead of rectangle areas.

Definition

The trapezoidal rule is a quadrature method that approximates the definite integral of a function \(f\) on \([a,b]\) by replacing the curve with the straight line joining \((a,f(a))\) and \((b,f(b))\).

\[\int_a^b f(x)\,dx \approx \frac{h}{2}\left[f(a)+2\sum_{i=1}^{n-1}f(x_i)+f(b)\right]\]

If \(f\) has a continuous second derivative on \([a,b]\), the error of the composite rule with step \(h=(b-a)/n\) is \(-\frac{(b-a)h^2}{12}f''(\xi)\) for some \(\xi\in(a,b)\), so the error is of order \(O(h^2)\). The rule is exact for polynomials of degree 1, since a line coincides with its trapezoid.

Worked example: for \(\int_0^1 x^2\,dx\) with \(n=2\) (\(h=0.5\)), we get \(\frac{0.5}{2}[0+2(0.25)+1]=0.375\), versus the exact value \(1/3\); the error \(0.0417\) matches the bound from the error term.

Typical error: applying the formula with \(n\) subintervals but using \(h=(b-a)/(n-1)\), or multiplying by \(h\) instead of \(h/2\) at the endpoints. Another common mistake is forgetting the factor 2 on the interior nodes.

Think about it

Why does the trapezoidal rule error depend on \(f''\) and not on \(f'\)?

Because the rule uses a straight line on each subinterval, and \(f''\) measures how far the curve departs from that line; if \(f''=0\), the error is zero.

Exercises 10

  1. What is the step \(h\) when applying the composite trapezoidal rule to \(\int_0^2 f(x)\,dx\) with \(n=4\) subintervals?
  2. What geometric shape does the trapezoidal rule use on each subinterval?
  3. What is the coefficient of \(f(x_i)\) for an interior node \(x_i\) in the composite trapezoidal rule?
  4. For which type of functions is the trapezoidal rule exact?
  5. What is the order of the error of the composite trapezoidal rule with respect to the step \(h\)?
  6. Apply the composite trapezoidal rule to \(\int_0^2 x^2\,dx\) with \(n=2\). What value do you get?
  7. Using the trapezoidal rule with \(n=4\) on \(\int_0^1 e^x\,dx\), what is the approximation given nodes \(f(0)=1\), \(f(0.25)\approx1.2840\), \(f(0.5)\approx1.6487\), \(f(0.75)\approx2.1170\), \(f(1)\approx2.7183\)?
  8. If the error of the composite trapezoidal rule with \(n=2\) on \(\int_0^1 x^2\,dx\) is \(0.0417\), what is the approximate error with \(n=4\)?
  9. What value does the trapezoidal rule give with \(n=1\) for \(\int_1^3 \frac{1}{x}\,dx\)?
  10. Let \(f\) satisfy \(|f''(x)|\le M\) on \([0,1]\). Show that the composite trapezoidal rule with \(n\) subintervals for \(\int_0^1 f\) has error bounded by \(\frac{M}{12n^2}\). Which statement is correct?

What comes next

References

  • Richard L. Burden, J. Douglas Faires, Annette M. Burden, Numerical Analysis, 10th edition, Cengage Learning, 2016.
  • Lloyd N. Trefethen, David Bau III, Numerical Linear Algebra, 1st edition, SIAM, 1997.