Question

[Choose the one correct answer and explain your reasoning.]

If the trapezium rule is used to estimate the integral \[\int^1_0 f(x)\, dx,\] by splitting the interval \(0 \le x \le 1\) into \(10\) intervals then an overestimate of the integral is produced. It follows that

  1. the trapezium rule with \(10\) intervals underestimates \(\int^1_0 2f(x)\; dx\);

  2. the trapezium rule with \(10\) intervals underestimates \(\int^1_0 (f(x)-1)\; dx\);

  3. the trapezium rule with \(10\) intervals underestimates \(\int^2_1 f(x-1)\; dx\);

  4. the trapezium rule with \(10\) intervals underestimates \(\int^1_0 (1-f(x))\; dx\).