Left Riemann Sum Calculator

Use the left edge of each slice for the height. Enter f(x), [a, b] and n to get Lₙ, every rectangle drawn and the table of left endpoints.

Function f(x)

Reads as

x^{2}

Try:

From aTo b

Subintervals n = 6

Left sum with n = 6

6.875

Exact $\int_{0}^{3} x^{2}\,dx = 9$ · error -2.125

The formula

L_{6} = \sum_{i=1}^{6} f\!\left(x_{i-1}\right)\Delta x,\quad \Delta x = \frac{3 - 0}{6} = 0.5

Each slice
$i$	sample $x$	height	area
0	0	0	0
1	0.5	0.25	0.125
2	1	1	0.5
3	1.5	2.25	1.125
4	2	4	2
5	2.5	6.25	3.125

All five methods, n = 6

Method	Estimate	Error
Left	6.875	-2.13
Right	11.375	2.38
Midpoint	8.9375	-0.0625
Trapezoid	9.125	0.125
Simpson	9	0

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An AI tutor reads the problem and steps above and explains them in plain words. It can make mistakes, so check it against the worked steps.

The left riemann sum formula

L_n = \Delta x\left[f(x_0) + f(x_1) + \cdots + f(x_{n-1})\right]

The left sum uses $x_0, x_1, \dots, x_{n-1}$ . It never uses the right end $b$ .

If $f$ is increasing on $[a,b]$ , each rectangle sits under the curve, so $L_n$ underestimates; if $f$ is decreasing, it overestimates.

Error: $|E_L| \le \frac{M(b-a)^2}{2n}$ where $M$ bounds $|f'|$ . The error shrinks only like $\frac1n$ .

Compare all five methods side by side on the Riemann sum calculator, or try another single rule: Trapezoidal rule, Simpson’s rule, Midpoint rule, Right Riemann sum.

Questions students ask

What is a left Riemann sum?

An area estimate using rectangles whose heights are the function values at the left endpoint of each subinterval: $L_n = \Delta x\sum_{i=0}^{n-1} f(x_i)$ .

Is a left Riemann sum an over- or underestimate?

An underestimate for increasing functions and an overestimate for decreasing functions.

How do you do a left Riemann sum from a table?

Multiply each width by the value at its left end and add, skipping the last table value.