Review question

# If the trapezium rule overestimates $\int^1_0 f(x)\, dx$, what can we deduce? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5324

## Suggestion

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 $\int^1_0 (1-f(x))\, dx$.

How is each function related to the original function $f(x)$? What transformation do we have to apply to $f(x)$ to get the new function each time?