Calculator Input
Formula Used
Step Size
h = (b - a) / n
Left and Right Riemann Sums
Left: ∫ f(x) dx ≈ h Σ f(a + i h), i = 0 to n-1
Right: ∫ f(x) dx ≈ h Σ f(a + i h), i = 1 to n
Midpoint Rule
∫ f(x) dx ≈ h Σ f(a + (i + 0.5) h), i = 0 to n-1
Trapezoidal Rule
∫ f(x) dx ≈ h[(f(a)+f(b))/2 + Σ f(a + i h)], i = 1 to n-1
Simpson's 1/3 Rule
∫ f(x) dx ≈ (h/3)[f(a)+f(b)+4Σf(odd nodes)+2Σf(even nodes)]
Simpson's 3/8 Rule
∫ f(x) dx ≈ (3h/8)[f(a)+f(b)+3Σf(non-multiples of 3)+2Σf(multiples of 3)]
How to Use This Calculator
- Enter a valid function in terms of
x, such asx^2+sin(x). - Provide the lower bound
aand upper boundb. - Choose the number of subintervals. Larger values often improve accuracy.
- Select one method or compare all supported methods together.
- Choose your preferred decimal precision.
- Click Approximate Integral to generate the result summary above the form.
- Export the generated output using the CSV or PDF buttons.
Example Data Table
Example function: f(x) = x^2, interval [0, 2], and n = 4.
| i | xi | f(xi) = xi2 |
|---|---|---|
| 0 | 0.00 | 0.0000 |
| 1 | 0.50 | 0.2500 |
| 2 | 1.00 | 1.0000 |
| 3 | 1.50 | 2.2500 |
| 4 | 2.00 | 4.0000 |
For this example, the exact integral is 8/3 ≈ 2.666667. Composite Simpson's 1/3 rule matches it exactly because the integrand is a quadratic polynomial.
Frequently Asked Questions
1. What does this calculator estimate?
It estimates a definite integral over a closed interval using common numerical methods. It is useful when an antiderivative is difficult, unavailable, or unnecessary for practical work.
2. Which method is usually most accurate?
For smooth functions, Simpson methods often outperform basic Riemann and trapezoidal estimates. Accuracy still depends on interval count, function behavior, and whether the rule's requirements are satisfied.
3. Why do Simpson methods sometimes fail?
Simpson's 1/3 rule requires an even number of subintervals. Simpson's 3/8 rule requires the count to be divisible by three. The calculator warns you when those conditions are not met.
4. Can I use trigonometric and logarithmic functions?
Yes. The calculator supports sin, cos, tan, inverse trig functions, square root, natural log, base-10 log, exponential, absolute value, pi, and e.
5. Why should I increase the number of subintervals?
More subintervals create a smaller step size, which usually improves approximation quality. Very oscillatory or singular-looking functions may still need careful interpretation and more refinement.
6. What is the reference integral shown in results?
It is a high-resolution Simpson estimate used as a comparison benchmark. It helps approximate absolute and relative error when an exact closed-form integral is not supplied.
7. Can this calculator handle reversed bounds?
Yes. If the lower bound is greater than the upper bound, the step size becomes negative and the resulting integral changes sign accordingly.
8. What do the CSV and PDF buttons export?
CSV exports the summary and node tables. PDF captures the displayed result section, including setup details, approximation results, and sampled values for documentation.