Right Riemann Sum Calculator

Q: How is the right Riemann sum used to define the integral?

Many textbooks define \int_a^b f(x)\,dx = \lim_{n\to\infty} R_n. Write R_n with x_i = a + i\frac{b-a}{n}, simplify with summation formulas, then take the limit.

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

Function f(x)

Reads as

x^{2}

Try:

From aTo b

Subintervals n = 6

Right sum with n = 6

11.375

Exact $\int_{0}^{3} x^{2}\,dx = 9$ · error 2.375

The formula

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

Each slice
$i$	sample $x$	height	area
0	0.5	0.25	0.125
1	1	1	0.5
2	1.5	2.25	1.125
3	2	4	2
4	2.5	6.25	3.125
5	3	9	4.5

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 right riemann sum formula

R_n = \Delta x\left[f(x_1) + f(x_2) + \cdots + f(x_n)\right]

The right sum uses $x_1, x_2, \dots, x_n$ . It never uses the left end $a$ .

If $f$ is increasing, $R_n$ overestimates; if decreasing, it underestimates. The gap between them is $R_n - L_n = \Delta x\,(f(b) - f(a))$ .

In sigma notation with $x_i = a + i\Delta x$ : $R_n = \sum_{i=1}^{n} f\!\left(a + i\tfrac{b-a}{n}\right)\tfrac{b-a}{n}$ , and its limit as $n \to \infty$ is the definite integral.

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

Questions students ask

What is a right Riemann sum?

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

Is a right Riemann sum an over- or underestimate?

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

How is the right Riemann sum used to define the integral?

Many textbooks define $\int_a^b f(x)\,dx = \lim_{n\to\infty} R_n$ . Write $R_n$ with $x_i = a + i\frac{b-a}{n}$ , simplify with summation formulas, then take the limit.