# Derivation of definite integral

Just a basic question to do with the derivation of the definite integral.

When using the areas of n rectangles to approximate the area under a given curve, why does the sum of the areas of n+1 rectangles give a more accurate approximation?

I can see it visually,I'm just not sure why it is so.

• This is unclear. More accurate than what? In general, the more rectangles the more accurate the approximation, at least for smooth functions (there may be pathological counterexamples). – Ethan Bolker Oct 24 '18 at 23:02
• But why exactly does the decrease in area caused by thinner rectangles outweigh the increase in area caused by an increase in the number of rectangles? – stochasticmrfox Oct 25 '18 at 20:18
• Why must more rectangles make more area? When there are more of them their bases are smaller. Thinner rectangles have smaller error in the variable height of the function. – Ethan Bolker Oct 25 '18 at 21:19

If the integrand has a bounded second derivative, the bound on the absolute error diminishes as $$\mathcal{O}(n^{-2})$$ as $$n \to \infty$$.
For any given function, the actual error for a given $$n$$ might fluctuate in a non-monotonic fashion as we increment $$n$$ by $$1$$. Nevertheless, the error is bounded by a function $$E(n)$$ that diminishes like $$1/n^2$$.