Accumulation Function Value Calculator

This calculator evaluates that total exactly for a polynomial represented by coefficients. Enter the coefficients in descending powers, the fixed lower bound, and the upper value where the accumulation function should be evaluated. The result includes the polynomial degree, antiderivative coefficients, endpoint evaluations, and final difference, making it useful for checking calculus homework, preparing examples, or validating deterministic calculations in software.

Enter the polynomial and both bounds correctly

Write the polynomial as an array of coefficients ordered from the highest power down to the constant term. For example, [3, 0, -2] represents 3x² - 2. The zero in the middle is important because it preserves the missing x term; omitting it would describe a different polynomial. A one-item array represents a constant function. Next, enter lower_bound as the fixed starting point of the accumulation and upper_bound as the variable endpoint at which you want its value. Both endpoints may be positive, negative, or equal, provided they are finite numbers. Unlike an interval-area tool, this calculator intentionally permits an upper bound below the lower bound. Reversing the direction changes the sign of the integral, while equal endpoints produce zero. Coefficients must also be finite numbers, and the bounded input accepts at most 1001 of them. Numeric coefficient arrays avoid ambiguity about multiplication, powers, parentheses, and missing terms, so check every position before interpreting the answer. The returned coefficients field repeats the accepted polynomial, allowing you to confirm that the intended degree and term order reached the calculation unchanged.

Follow the analytic accumulation calculation

For an accumulation function A(x) = ∫ from a to x of p(t) dt, the calculator first constructs a convenient antiderivative F of the polynomial. Each term c·tⁿ becomes c divided by n + 1, multiplied by t raised to n + 1. A zero constant is appended because any antiderivative constant cancels in the endpoint subtraction. The antiderivative_coefficients output lists this polynomial in descending powers, including that final zero. The algorithm then evaluates F at upper_bound and lower_bound using a bounded Horner-style pass and subtracts the latter from the former. Thus value equals antiderivative_at_upper minus antiderivative_at_lower. This is the Fundamental Theorem of Calculus applied directly, with no sampling or numerical quadrature. It is exact as an algebraic method, although displayed decimal values use finite-precision JavaScript numbers and are normalized to stable significant digits. If intermediate powers overflow the supported numeric range, the request is rejected instead of returning an infinity or a misleading partial result. You can reproduce the calculation by integrating each term, evaluating both endpoints, and checking the reported subtraction.

Interpret sign, units, and the variable endpoint

The returned value is signed accumulation, not automatically geometric area. Where the polynomial is below the horizontal axis, its contribution is negative; where it is above, its contribution is positive. These effects may cancel. If you need total geometric area, split the interval at every real zero and add the absolute value of each separate integral. The fixed lower bound anchors a whole accumulation function, while upper_bound selects one value of that function. To study several points, keep coefficients and lower_bound unchanged and evaluate multiple upper bounds. The derivative of the resulting accumulation function is the original polynomial, so the result also supports a useful consistency check: A increases where p is positive and decreases where p is negative. Units multiply as they do for every definite integral. If p measures a rate in units per second and the bounds are seconds, the accumulation value is measured in units. Reversing the endpoints negates the result, and setting them equal returns zero because nothing is accumulated over a zero-length interval. The browser calculation is available without charge, while an API request uses the published base price of $0.002.