Integral Calculator

How to Use the Integral Calculator

Enter the function you want to integrate using x as the variable, then set the lower bound a and upper bound b of the interval. The calculator computes the definite integral ∫_a^b f(x) dx in real time as you type. You can use standard mathematical notation including parentheses, exponents with the caret operator (x^2), and built-in functions like sin(x), exp(x), and ln(x). The constants pi and e are also recognised, so you can write expressions like sin(pi*x) or exp(-x^2) directly.

The result represents the signed area between the curve f(x) and the x-axis over the interval. Areas above the x-axis count as positive and areas below count as negative. If you swap the bounds (enter a value for a that is larger than b), the calculator automatically flips the sign of the result, matching the standard mathematical convention.

How Simpson’s Rule Works

This calculator uses Simpson’s rule, a numerical integration method that approximates the area under a curve by fitting parabolic arcs through groups of three points. The interval [a, b] is divided into n equal subintervals (n must be even), and the integral is approximated by the formula:

∫_a^b f(x) dx ≈ (h/3) · [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 4f(x_n-1) + f(x_n)]

where h = (b − a) / n is the width of each subinterval. Simpson’s rule is exact for polynomials up to degree three, and for smoother functions it converges much faster than the simpler trapezoidal rule. With the default 1000 intervals, accuracy is typically better than one part in a million for well-behaved functions.

Supported Functions and Operators

The expression parser supports the four basic operators (+, −, *, /), exponentiation with ^, unary minus, and parentheses for grouping. The following built-in functions are available: sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, ln (natural logarithm), log10 (base-10 logarithm), exp, sqrt, and abs. The constants pi (π ≈ 3.14159) and e (≈ 2.71828) can be used anywhere a number is expected.

When Numerical Integration Falls Short

Numerical methods work best on smooth, continuous functions over finite intervals. If your function has a vertical asymptote inside the interval (such as 1/x crossing zero), Simpson’s rule will produce nonsense or an error. Improper integrals with infinite bounds are also not supported directly — you would need to truncate the interval to a large but finite range and accept the resulting approximation. For symbolic answers (such as obtaining x³/3 + C as the antiderivative of x²), you need a computer algebra system rather than a numerical calculator like this one.

Common Use Cases

Definite integrals appear constantly in physics (work done by a variable force, displacement from a velocity function), engineering (signal energy, fluid flow), statistics (cumulative probabilities of continuous distributions), and economics (consumer surplus, present value of continuous cash flows). Calculus students use them to compute areas, volumes of revolution, arc lengths, and centres of mass. This calculator gives you a quick way to check homework answers, validate hand calculations, or explore how a function behaves over a region without setting up a full mathematical software environment.

Frequently Asked Questions

What is a definite integral?

A definite integral computes the signed area under the curve of a function f(x) between two bounds a and b. It is written as ∫_a^b f(x) dx and represents the accumulation of the function’s values over that interval.

What method does this calculator use?

This calculator uses Simpson’s rule, a numerical integration method that fits parabolas through pairs of intervals. It is exact for polynomials up to degree three and converges very quickly for smooth functions.

Which functions and operators are supported?

You can use +, −, *, /, ^ for power, parentheses for grouping, and the variable x. Built-in functions include sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, ln, log10, exp, sqrt, and abs. The constants pi and e are also recognised.

Why is the result not exact for some functions?

Simpson’s rule is a numerical approximation. For polynomials up to degree three the result is exact, but for other functions there is a small error that depends on how many intervals you use. Increasing the intervals reduces the error.

Can this calculator handle improper integrals?

Not reliably. If the function is undefined at a bound or has a vertical asymptote inside the interval, the calculator will return an error or an inaccurate result. Restrict the bounds to a region where the function is finite and continuous.