Midpoint Rule Integral Calculator

Approximate integrals from expressions using reliable midpoint sampling. Inspect intervals, values, and convergence details quickly. Export results, review examples, and understand each calculation step.

Calculator Form

Function f(x)

Use x, pi, e, +, -, *, /, ^, parentheses, and functions like sin(x), cos(x), sqrt(x), abs(x), log(x), ln(x).

Lower Bound a

Upper Bound b

Subintervals n

Decimal Places

Graph Points

Angle Mode

Example Data Table

This static example uses f(x) = x^2 on [0, 2] with n = 4.

i	Left	Right	Midpoint	f(Midpoint)	Contribution
1	0.0000	0.5000	0.2500	0.0625	0.0313
2	0.5000	1.0000	0.7500	0.5625	0.2813
3	1.0000	1.5000	1.2500	1.5625	0.7813
4	1.5000	2.0000	1.7500	3.0625	1.5313
Midpoint Rule Estimate					2.6250

Formula Used

Step size: h = (b - a) / n

Midpoint of each interval: m_i = a + (i - 0.5)h

Midpoint rule estimate: M_n = h × Σ f(m_i) for i = 1 to n

Average value on the interval: f_avg = M_n / (b - a)

Theoretical midpoint error: E = -((b - a)³ / (24n²)) f''(ξ)

Practical error estimate used here: |M_2n - M_n| / 3

The calculator also shows a refined midpoint estimate with 2n subintervals. That comparison helps you judge whether your chosen partition count is already stable enough.

How to Use This Calculator

Enter the function using x as the variable.
Type the lower and upper integration bounds.
Choose the number of subintervals n.
Select decimal places and graph resolution.
Choose radian or degree mode for trigonometric work.
Press the calculate button to show the result above the form.
Review the summary, table, and graph.
Use the export buttons when you need a file copy.

FAQs

1. What does the midpoint rule estimate?

It approximates a definite integral by sampling the function at each subinterval midpoint and multiplying the sum by the common interval width.

2. How many subintervals should I use?

Start with a modest value like 8 or 16. Increase n until the refined midpoint estimate changes very little. Smoother functions usually converge faster.

3. What function syntax works here?

Use x, numbers, parentheses, pi, e, powers with ^, and standard functions such as sin, cos, tan, sqrt, abs, exp, log, and ln.

4. Can I integrate trigonometric expressions?

Yes. Select radian mode for standard calculus work. Degree mode is available if your expression values are intended to be angle-based in degrees.

5. Why is there also a Simpson estimate?

It acts as a reference value for comparison. It helps you see whether your midpoint approximation is close to a finer numerical estimate.

6. Does the calculator allow reversed bounds?

Yes. If the upper bound is smaller than the lower bound, the step size becomes negative and the integral estimate reflects the reversed direction.

7. What if the function has gaps or undefined points?

If the expression becomes invalid at sampled positions, the calculator shows a message. Functions with singularities may require interval splitting or another method.

8. Can I export the output?

Yes. Use the CSV button for spreadsheet style data and the PDF button for a compact report containing the main results and table rows.