Calculator inputs
Formula used
Signed area: \(\int_a^b f(x)\,dx\) measures net area, so regions below the x-axis reduce the total.
Geometric area: \(\int_a^b |f(x)|\,dx\) treats every shaded region as positive magnitude.
Trapezoidal Rule: \(\frac{h}{2}[f(x_0)+2\sum f(x_i)+f(x_n)]\)
Midpoint Rule: \(h\sum f(m_i)\), where each midpoint \(m_i\) represents one rectangle.
Simpson's Rule: \(\frac{h}{3}[f(x_0)+4\sum f(x_{odd})+2\sum f(x_{even})+f(x_n)]\)
Average height equals signed area divided by width, \((b-a)\), giving an equivalent constant graph height.
How to use this calculator
- Enter a valid function using x as the variable.
- Set lower and upper bounds for the integration interval.
- Choose Simpson, trapezoidal, or midpoint estimation.
- Adjust the subinterval count for the detail level.
- Choose graph sample points and visible decimal precision.
- Enable geometric area mode when you want absolute magnitude.
- Press Calculate Area to show results above the form.
- Use the CSV or PDF buttons to save the current output.
Example data table
| Function | Lower | Upper | Method | Intervals | Signed Area | Geometric Area |
|---|---|---|---|---|---|---|
| x^2 | 0 | 3 | Simpson's Rule | 12 | 9.000000 | 9.000000 |
| sin(x) | 0 | 3.141593 | Trapezoidal Rule | 24 | 1.997144 | 1.997144 |
| x^3-4x | -2 | 2 | Midpoint Rule | 40 | 0.000000 | 8.006400 |
Current interval preview
Run a calculation to preview sampled nodes, function values, and integration weights used by the selected method.
Frequently asked questions
1. What does this calculator measure?
It estimates the area related to a curve between two x-values. You can view net signed area or switch to geometric area, which counts every shaded region as positive.
2. When should I use geometric area mode?
Use geometric area when you want the full shaded magnitude regardless of whether the curve falls below the x-axis. This is common in geometry, materials work, and teaching examples.
3. Why does Simpson's Rule change my interval count?
Simpson's Rule requires an even number of subintervals. If you enter an odd value, the calculator increases it by one so the formula remains valid.
4. Which method is usually most accurate?
For smooth curves, Simpson's Rule often performs best with the same interval count. Trapezoidal and midpoint methods are still valuable for comparison, learning, or simpler estimate workflows.
5. What functions can I enter?
You can enter expressions using x, numbers, parentheses, powers, and common functions such as sin, cos, tan, sqrt, abs, ln, log, and exp.
6. Why can the signed area be zero?
Signed area adds positive and negative contributions together. Symmetric or crossing curves may cancel out, producing a result near zero even when visible shaded regions exist.
7. What does the refinement change indicate?
It compares your current estimate against the same method with twice as many intervals. A smaller difference usually suggests the current result is becoming more stable.
8. Do the export buttons save the latest result?
Yes. The CSV and PDF buttons resubmit the current calculator settings, recalculate the result, and download a clean summary you can store or share.