Integral Calculator

This calculator finds the exact definite integral of a chosen function family — polynomial, trigonometric, or exponential — between a Lower and Upper Bound, using the known antiderivative formula for each family rather than a numerical approximation. The Upper Bound has by far the biggest pull on the result of the four movable numeric inputs, well ahead of the Coefficient and the Exponent/Parameter — though Function Type itself moves the result more than any single numeric input does, since switching between polynomial, trig, and exponential swaps the entire antiderivative formula, not just adjusting a coefficient inside a fixed one. That also means the Exponent's direction of pull is not fixed across function types: at the polynomial defaults raising the Exponent lowers the integral (explained below), but for the trig function the same nudge can raise it instead, since sine and its antiderivative flip sign as the frequency parameter sweeps past different multiples of the bounds — a fact about which formula is active, not about the Exponent on its own.

At the calculator's polynomial defaults, the Upper Bound defaults to exactly 1, and 1 raised to any power still returns 1 — so Upper Bound raised to (Exponent plus 1) collapses to exactly 1 and the integral reduces to Coefficient divided by (Exponent plus 1) at these particular settings, meaning the Exponent's own influence here is really just about shrinking or growing that one denominator, not about a changing upper limit of integration the way it would at any other bound. That's also why raising the Exponent at these defaults actually lowers the result rather than raising it: a bigger exponent means dividing the same numerator by a bigger number. The Absolute Area output differs from the signed Definite Integral whenever part of the curve dips below the x-axis across the interval — the definite integral lets those negative and positive regions cancel out, while Absolute Area sums the magnitude of every region regardless of sign, computed separately via a numerical Simpson's-rule pass over the absolute value of the function.