A definite integral measures accumulated signed area. Before taking a limit, replace the curve by rectangles whose areas are easy to add.

Divide \([a,b]\) into \(n\) equal subintervals. If \(x_r=a+r\Delta x\), then

\[\Delta x=\frac{b-a}{n},\qquad \text{rectangle area}\approx \mathrm{f}(\text{sample point})\,\Delta x.\]

Use the left endpoint of each subinterval for \(L_n\), or the right endpoint for \(R_n\). The width is the same; only the sampled height changes.

\[L_n=\Delta x\sum_{r=0}^{n-1}\mathrm{f}(x_r),\qquad R_n=\Delta x\sum_{r=1}^{n}\mathrm{f}(x_r).\]

\[L_n=\frac1n\sum_{r=0}^{n-1}\mathrm{f}\!\left(\frac rn\right)=\frac1n\sum_{r=1}^{n}\mathrm{f}\!\left(\frac{r-1}{n}\right),\qquad R_n=\frac1n\sum_{r=1}^{n}\mathrm{f}\!\left(\frac rn\right).\]

For the positive curves shown, compare each rectangle with the curve over the same subinterval.

As \(n\) increases, the rectangles become thinner and the gaps shrink. For an integrable function, both endpoint sums approach the same value.

\[\lim_{n\to\infty}L_n=\lim_{n\to\infty}R_n=\int_a^b \mathrm{f}(x)\,\mathrm{d}x.\] 

A Riemann sum is a finite approximation; the definite integral is its limiting value.

Default solution

For \(n\) equal subintervals of \([0,1]\), each rectangle has width \(\frac1n\). A left Riemann sum is \(L_n=\frac1n\left[f(0)+f\left(\frac1n\right)+\cdots+f\left(\frac{n-1}{n}\right)\right]=\frac1n\sum_{r=0}^{n-1}f\left(\frac rn\right)\).

A right Riemann sum is \(R_n=\frac1n\left[f\left(\frac1n\right)+f\left(\frac2n\right)+\cdots+f(1)\right]=\frac1n\sum_{r=1}^{n}f\left(\frac rn\right)\).

For an increasing function, \(L_n\) is an underestimate and \(R_n\) is an overestimate. For a decreasing function, \(L_n\) is an overestimate and \(R_n\) is an underestimate.

As \(n\to\infty\), both estimates tend to the exact area: \(\lim_{n\to\infty}L_n=\lim_{n\to\infty}R_n=\int_0^1 f(x)\,\mathrm{d}x\).

Behaviour of \(\mathrm{f}\)	Left sum	Right sum
Increasing	Underestimate	Overestimate
Decreasing	Overestimate	Underestimate

Riemann Sums and Bounds

Riemann Sums and Bounds

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