Midpoint Rule Calculator

Q: How do you do the midpoint rule?

Find \Delta x = \frac{b-a}{n}, list the midpoints m_i = a + (i - \tfrac12)\Delta x, evaluate f at each, add them and multiply by \Delta x.

Q: What is the error bound for the midpoint rule?

|E_M| \le \frac{K(b-a)^3}{24n^2} where K bounds |f''| on [a,b].

Q: Is the midpoint rule an over- or underestimate?

An underestimate where f is concave up and an overestimate where it is concave down.

Sample each slice at its centre. Enter f(x), [a, b] and n to get Mₙ with every rectangle drawn, the midpoint table and the error compared with the exact integral.

Function f(x)

Reads as

x^{2}

Try:

From aTo b

Subintervals n = 6

Midpoint sum with n = 6

8.9375

Exact $\int_{0}^{3} x^{2}\,dx = 9$ · error -0.0625

The formula

M_{6} = \sum_{i=1}^{6} f\!\left(\tfrac{x_{i-1}+x_i}{2}\right)\Delta x,\quad \Delta x = \frac{3 - 0}{6} = 0.5

Each slice
$i$	sample $x$	height	area
0	0.25	0.0625	0.03125
1	0.75	0.5625	0.28125
2	1.25	1.5625	0.78125
3	1.75	3.0625	1.53125
4	2.25	5.0625	2.53125
5	2.75	7.5625	3.78125

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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The midpoint rule formula

M_n = \Delta x\left[f(m_1) + f(m_2) + \cdots + f(m_n)\right],\quad m_i = \frac{x_{i-1} + x_i}{2}

The midpoint rule is a Riemann sum whose sample point is the centre of each slice. On a slope, the bit of rectangle that sticks out above the curve on one side nearly cancels the gap on the other, which is why it beats left and right sums.

Error bound: if $|f''(x)| \le K$ on $[a,b]$ , then $|E_M| \le \frac{K(b-a)^3}{24n^2}$ , half the trapezoidal bound.

For a concave-up function the midpoint rule underestimates; for concave-down it overestimates (the opposite of the trapezoidal rule).

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

Questions students ask

How do you do the midpoint rule?

Find $\Delta x = \frac{b-a}{n}$ , list the midpoints $m_i = a + (i - \tfrac12)\Delta x$ , evaluate $f$ at each, add them and multiply by $\Delta x$ .

What is the error bound for the midpoint rule?

$|E_M| \le \frac{K(b-a)^3}{24n^2}$ where $K$ bounds $|f''|$ on $[a,b]$ .

Is the midpoint rule an over- or underestimate?

An underestimate where $f$ is concave up and an overestimate where it is concave down.

Midpoint rule vs trapezoidal rule: which is better?

For smooth functions the midpoint rule is about twice as accurate, and it needs one fewer function evaluation.