Integral Calculator

A definite integral is one number that measures the signed area between a curve and the x axis, from a lower limit a to an upper limit b. Where the curve sits above the axis, the area counts as positive. Where it dips below the axis, the area counts as negative. The integral is the net total after the two cancel out.

The same number answers accumulation questions in everyday life. If the curve shows velocity over time, the integral gives total distance. With real units on the axes, the area becomes a real quantity.

This calculator takes your function and limits and estimates the integral with the composite Simpson rule, fitting parabolas through groups of three sample points and adding their areas. The breakdown shows every sample point used.

The Fundamental Theorem of Calculus, Part 2, connects the integral to the antiderivative. If F is any function with F'(x) = f(x), then the integral from a to b of f(x) dx equals F(b) minus F(a). In symbols this is written as integral_a^b f(x) dx = F(b) - F(a).

Each symbol has one job. f(x) is the integrand, the function being integrated. x is the variable. dx says the integration runs along x. a is the lower limit and b is the upper limit. F is the antiderivative, and F(b) - F(a) is the net change of F across the interval.

The calculator uses the composite Simpson rule, which reads integral approx (h/3) [f(x0) + 4 f(x1) + 2 f(x2) + 4 f(x3) + ... + f(xn)], with h = (b - a) / n and n even. It doubles the subinterval count from 8 until the value settles within 1e-10. Simpson's rule is exact for polynomials up to degree 3 and highly accurate for smooth functions. For quick checks of trig values like sin(x), the trigonometry calculator is a handy companion.

Find the integral of x^2 from 0 to 3. The antiderivative is x^3 / 3, so the exact answer is 27/3 - 0 = 9. Here is the calculator's path with Simpson's rule and 8 subintervals.

1. h = (3 - 0) / 8 = 0.375.
2. Sample points: 0, 0.375, 0.75, 1.125, 1.5, 1.875, 2.25, 2.625, 3.
3. Function values: 0, 0.140625, 0.5625, 1.265625, 2.25, 3.515625, 5.0625, 6.890625, 9.
4. Weighted sum: (0 + 9) + 4 x 11.8125 + 2 x 7.875 = 72.
5. Multiply by h/3 = 0.125. The result is 0.125 x 72 = 9.

The calculator then refines with more subintervals until the value stops changing, confirming 9. If you want to check the arithmetic of any step by hand, the scientific calculator handles the squaring and summing without rounding errors.

1. Type the function in the Integrand field, for example x^2, sin(x), or e^x. Use ^ for powers, sqrt(), sin(), cos(), tan(), ln(), exp(), and the constants pi and e.
2. Choose the variable if your expression uses t or u instead of x.
3. Enter the lower limit a and the upper limit b. Each accepts a number or an expression like pi/2.
4. Press Calculate. The answer appears first, then the step-by-step breakdown.
5. Press Reset to clear all fields and start over.

A definite integral can be positive, negative, or zero. Every part of the curve above the x axis adds to the total, and every part below it subtracts.

A full wave of sin(x) from 0 to 2pi integrates to 0. The hump above the axis and the hump below it cancel exactly. That zero is the correct signed answer, not an error.

Reversed limits flip the sign, so the integral of x^2 from 3 to 0 is -9. When both limits are equal, the integral is 0. The exponent calculator is useful for checking individual terms by hand.

The most common mistake is entering the limits in the wrong order and missing the sign change. Check which limit is lower first.

The second is asking for an integral that does not exist as a finite number. The integral of 1/x from -1 to 1 crosses a singularity at 0, so it has no finite value. The calculator detects this and warns you instead.

The third is a typo in the expression. A missing parenthesis or a misspelled function name stops the parser, and the error message points at the exact spot. Fix that spot and calculate again.