Riemann Sums

Applications of Intergration

International Baccalaureate Higher Level Mathematics

IB HL Math

**Applications of Integration - Riemann Sums**

**Left Riemann Sums:**
• For increasing function: Underestimate
• For decreasing function: Overestimate

Estimated Area \(= \displaystyle\frac{1}{n}\left[f(0) + f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + ... + f\left(\frac{n-1}{n}\right)\right]\)
\(= \displaystyle\frac{1}{n}\sum_{r=0}^{n-1} f\left(\frac{r}{n}\right)\)

**Right Riemann Sums:**
• For increasing function: Overestimate
• For decreasing function: Underestimate

Estimated Area \(= \displaystyle\frac{1}{n}\left[f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + ... + f\left(\frac{n}{n}\right) + f(1)\right]\)
\(= \displaystyle\frac{1}{n}\sum_{r=1}^{n} f\left(\frac{r}{n}\right)\)

As \(n \rightarrow \infty\), estimated area \(\rightarrow \displaystyle\displaystyle\int\nolimits_0^1 \mathrm{f}(x) \, \mathrm{d}x\)

i.e. \(\displaystyle\lim_{n \rightarrow \infty} \frac{1}{n}\sum f\left(\frac{r}{n}\right) = \displaystyle\int\nolimits_0^1 \mathrm{f}(x) \, \mathrm{d}x\)

(Estimation from both Left and Right Riemann Sum)

Riemann Sums

Riemann Sums

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