Calculator Inputs
Use function mode for an equation or table mode for known data pairs.
Example Data Table
This example approximates ∫04 x² dx using four equal subintervals.
| Point | x | y = x² | Weight | Weighted y |
|---|---|---|---|---|
| 0 | 0 | 0 | 0.5 | 0.0 |
| 1 | 1 | 1 | 1.0 | 1.0 |
| 2 | 2 | 4 | 1.0 | 4.0 |
| 3 | 3 | 9 | 1.0 | 9.0 |
| 4 | 4 | 16 | 0.5 | 8.0 |
Here, h = 1 and the approximation becomes 1 × (0 + 1 + 4 + 9 + 8) = 22.
Formula Used
Composite trapezoidal rule for equal spacing:
T = h [(f(x₀) + f(xₙ)) / 2 + Σ f(xᵢ)] for interior points i = 1 to n - 1.
Step size: h = (b - a) / n
Segment form for uneven table spacing:
T = Σ [(xᵢ₊₁ - xᵢ) × (yᵢ + yᵢ₊₁) / 2]
If you provide an exact value, the calculator also reports absolute and relative error for the approximation.
How to Use This Calculator
- Select Function Expression to integrate a typed formula, or Table Data to integrate known points.
- Enter the required values. In function mode, provide f(x), lower limit, upper limit, and subinterval count.
- In table mode, paste one x, y pair per line in increasing x order.
- Optionally enter the exact integral value to compare the approximation against a known answer.
- Choose your decimal precision and click Calculate.
- Review the summary, point table, interval contributions, and optional error values shown above the form.
- Use the CSV and PDF buttons to save the results for reporting or revision.
Frequently Asked Questions
1. What does the trapezoidal rule estimate?
It estimates the value of a definite integral by replacing curved regions with trapezoids. The summed trapezoid areas approximate the area under the curve.
2. When should I use function mode?
Use function mode when you know the equation of the curve. The calculator samples evenly spaced x-values between the lower and upper limits.
3. When is table mode better?
Table mode is best when you have measured or observed data points instead of a formula. It can also handle uneven spacing between x-values.
4. Does increasing subintervals improve accuracy?
Usually yes. More subintervals create narrower trapezoids, which often follow the curve better and reduce approximation error for smooth functions.
5. What functions are supported in the expression field?
You can use x, parentheses, powers, and common functions such as sin, cos, tan, sqrt, exp, log, log10, abs, floor, and ceil.
6. Why would I enter an exact integral value?
An exact value helps you compare the approximation against a trusted answer. The calculator then reports absolute error and relative percentage error.
7. Can the calculator work with decreasing limits?
Yes. If the upper limit is less than the lower limit, the step size becomes negative and the resulting integral sign changes accordingly.
8. What causes invalid result messages?
Errors appear when the expression syntax is unsupported, x-values are not increasing, intervals are missing, or the formula produces undefined numeric output.