Calculator Form
Supported functions include sin, cos, tan, sqrt, abs, log, ln, and exp.
Example Data Table
| Property | Function Setup | Bounds | Expected Result |
|---|---|---|---|
| Direct value | f(x) = x^2 | a = 0, b = 3 | 9.000000 |
| Additivity | f(x) = sin(x), c = pi/2 | a = 0, b = pi | Full integral equals both parts combined |
| Even symmetry | f(x) = x^2 | a = -3, b = 3 | 18.000000 |
| Odd symmetry | f(x) = x^3 | a = -2, b = 2 | 0.000000 |
Formula Used
The direct value uses the definite integral definition: ∫ab f(x) dx.
Reversal of limits uses ∫ab f(x) dx = -∫ba f(x) dx.
Additivity uses ∫ab f(x) dx = ∫ac f(x) dx + ∫cb f(x) dx.
Linearity uses ∫ab [k f(x)] dx = k∫ab f(x) dx and ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx.
Symmetry uses ∫-aa f(x) dx = 2∫0a f(x) dx for even functions and ∫-aa f(x) dx = 0 for odd functions.
This calculator estimates each integral numerically with Simpson’s Rule: ∫ab f(x) dx ≈ h/3 [f(x0) + 4f(x1) + 2f(x2) + ... + f(xn)].
How to Use This Calculator
- Choose the definite integral property you want to test.
- Enter f(x) and optionally g(x) for sum or difference rules.
- Provide bounds a and b, plus split point c when needed.
- Enter scalar k for linearity checks.
- Set integration steps and decimal places.
- Press Submit to show the result below the header and above the form.
- Download the computed results or example table as CSV or PDF when needed.
Frequently Asked Questions
1. What does this calculator measure?
It numerically evaluates definite integrals and checks common properties such as reversal, additivity, linearity, and symmetry over chosen bounds.
2. Which functions can I enter?
You can enter algebraic and common transcendental expressions using x, numbers, parentheses, powers, and functions like sin, cos, tan, sqrt, abs, log, ln, and exp.
3. Why do symmetry checks need matching limits?
Even and odd symmetry properties apply on intervals of the form [-a, a]. If the bounds are not symmetric, the property may not hold.
4. What is the split point c used for?
The split point divides one interval into two adjacent intervals. The additivity property compares the full integral with the sum of both smaller integrals.
5. Why can two values differ slightly?
This page uses numerical integration, not symbolic antiderivatives. Small differences can appear because Simpson’s Rule estimates the area using many subintervals.
6. How many integration steps should I use?
Higher step counts usually improve accuracy for smooth functions. Start around 400, then increase if the verification difference is larger than expected.
7. Can I test the sum and difference rules?
Yes. Enter both f(x) and g(x), choose the required rule, and the calculator compares the combined integral with the sum or difference of separate integrals.
8. Does the result represent geometric area only?
Not always. A definite integral returns signed area, so sections below the x-axis contribute negative values unless a separate absolute-area method is used.