Definite Integral Function Calculator

Calculator inputs

Function f(x)

Supported: sin, cos, tan, exp, ln, log, log10, sqrt, abs, sec, csc, cot, pi, e.

Lower bound a

Upper bound b

Segments

Higher values usually improve accuracy.

Method

Decimal places

Use absolute area instead of signed area

Reset

Quick examples

Formula used

Exact definite integral idea

If an antiderivative F(x) is known, then the exact value is ∫_a^b f(x) dx = F(b) - F(a). This calculator estimates the value numerically.

Composite trapezoidal rule

Let h = (b - a) / n. Then ∫_a^b f(x) dx ≈ h[½f(x₀) + f(x₁) + ... + f(x_n-1) + ½f(x_n)].

Composite midpoint rule

The midpoint estimate is ∫_a^b f(x) dx ≈ h Σ f(a + (i + ½)h), where i = 0 to n - 1.

Composite Simpson's Rule

For even n, ∫_a^b f(x) dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + ... + 4f(x_n-1) + f(x_n)].

How to use this calculator

Enter a function using x as the variable, such as sin(x), x^2 + 3*x, or exp(-x^2).
Type the lower bound a and upper bound b.
Choose the number of segments. Larger counts usually give better estimates.
Select a numerical method, or choose Compare All Methods to review all three together.
Set the decimal places you want in the output.
Enable absolute area if you want total area without cancellation between positive and negative parts.
Press Calculate Integral to show the result directly below the header and above the form.
Use the CSV and PDF buttons to export the summary, comparison table, and sample points.

Example data table

Function	Interval	Segments	Method	Estimated integral	Notes
sin(x)	[0, 3.141593]	100	Simpson's Rule	2.000000	Smooth periodic function with strong convergence.
x^2	[0, 3]	120	Simpson's Rule	9.000000	Polynomial estimates are typically very stable.
sqrt(x)	[0, 4]	120	Trapezoidal Rule	5.333023	Useful for endpoint-heavy sampling.
exp(-x^2)	[0, 2]	200	Midpoint Rule	0.882081	Good quick estimate for smooth decay.

FAQs

1. What functions can I enter?

You can enter expressions in x using operators, parentheses, constants pi and e, and functions such as sin, cos, tan, exp, ln, log, log10, sqrt, abs, sec, csc, and cot.

2. Why does Simpson's Rule change my segment count?

Simpson's Rule needs an even number of subintervals. When you enter an odd count, the calculator adds one segment automatically and shows a note beside the result.

3. Does this calculator return exact answers?

No. This tool uses numerical approximation. Accuracy usually improves when the function is smooth and the segment count is higher. Some simple functions can still match the exact value very closely.

4. What happens if I reverse the bounds?

The calculator keeps the order you enter. Reversing the bounds changes the sign of the definite integral, which follows standard calculus rules.

5. When should I use the midpoint rule?

Midpoint is useful for quick estimates and often beats the trapezoidal rule on curved functions. It also avoids endpoint evaluation, which helps when endpoints are awkward.

6. What does absolute area mode mean?

Absolute area converts sampled function values to magnitudes before summing. Use it when you want total area without positive and negative regions canceling each other.

7. Why am I seeing an invalid function error?

Unsupported words, missing parentheses, division by zero, or undefined values can trigger errors. Check spelling, use x as the variable, and avoid formulas outside the entered interval.

8. How can I improve precision?

Increase the segment count first. Then compare Simpson, midpoint, and trapezoidal estimates. When the values converge closely, your approximation is usually more reliable.