It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of $\int_{a}^{b}f(x)dx$.

I can also see that the converse does not necessarily hold. That is,

If one application of the trapezoidal rule over $a\leq x \leq b$ yields the exact value of $\int_{a}^{b}f(x)dx$, then it is not necessarily the case that $f(x)$ has constant gradient.

Any function that is odd about the inflexion point would act as counter examples to demonstrate this.

My question is, are such functions the ONLY non-linear functions whose integrals over $a\leq x \leq b$ can be found using a single trapezium over the same domain?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.