Let $A$ be the exact area over $[a,b]$ under $y=f(x)$.
If $f(x) \geq 0$ (positive), and increasing, then $\forall x \in [a,b]$, Left Riemann Sum $\leq$ A $\leq$ Right Riemann Sum.
How do I prove this? I don't know where to start
2026-03-28 08:36:27.1774686987
How to prove Left Riemann Sum is underestimate and Right Riemann sum is overestimate?
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Well, for a single interval and nondecreasing $f$:
$$a\le x\le b\implies f(a)\le f(x) \le f(b) \implies \int_a^b f(a)\,dx\le \int_a^bf(x)\,dx \le \int_a^bf(b)\,dx$$
$$\implies (b-a)f(a) \le \int_a^bf(x)\,dx \le (b-a)f(b)$$
The general case comes from adding this inequality up across intervals.