The graph \(y=f(x)\) of a function is drawn below for \(0 \le x \le 1\).

a roughly parabolic graph in four straight line parts

The trapezium rule is then used to estimate

\[\displaystyle\int_0^1 \! f(x) \, \mathrm{d}x\]

by dividing \(0 \le x \le 1\) into \(n\) equal intervals. The estimate calculated will equal the actual integral when

  1. \(n\) is a multiple of \(4\);

  2. \(n\) is a multiple of \(6\);

  3. \(n\) is a multiple of \(8\);

  4. \(n\) is a multiple of \(12\).

The notation \[\displaystyle\int_0^1 \! f(x) \, \mathrm{d}x\] means the area between the \(f(x)\) and the \(x\)-axis, from \(x=0\) to \(x=1\).