Definite Integral Calculator
Computes the exact area under the curve y = axⁿ + c between two x-values using the Fundamental Theorem of Calculus. Perfect for students solving area, displacement, or accumulation problems.
About this calculator
The definite integral of f(x) = axⁿ + c from x = L to x = U is computed using the antiderivative F(x) = a·x^(n+1)/(n+1) + c·x, evaluated at both bounds. By the Fundamental Theorem of Calculus: ∫[L to U] (axⁿ + c) dx = F(U) − F(L). Expanding this gives: [a/(n+1)] × (U^(n+1) − L^(n+1)) + c × (U − L). The formula used here applies exactly that expression. Note that this formula is undefined when n = −1 (which would require a logarithmic antiderivative). The result represents the signed area between the curve and the x-axis over the interval [L, U], with regions below the axis contributing negative area.
How to use
Integrate f(x) = 3x² + 4 from x = 1 to x = 3. Enter coefficient = 3, power = 2, lower = 1, upper = 3, constant = 4. Compute: [3/(2+1)] × (3³ − 1³) + 4 × (3 − 1) = [3/3] × (27 − 1) + 4 × 2 = 1 × 26 + 8 = 34. The area under y = 3x² + 4 between x = 1 and x = 3 is exactly 34 square units.
Frequently asked questions
What is the difference between a definite and an indefinite integral?
An indefinite integral produces a family of antiderivatives written as F(x) + C, where C is an arbitrary constant. A definite integral evaluates that antiderivative at two specific bounds and subtracts: ∫[L to U] f(x) dx = F(U) − F(L), yielding a single numerical value. The constant C cancels out in this subtraction, which is why you do not need to worry about it when computing definite integrals. The definite integral represents a concrete quantity such as area, displacement, or accumulated change.
Why does the definite integral formula fail when the power equals negative one?
When n = −1, the antiderivative of x⁻¹ is ln|x|, not x⁰/0, which would be division by zero. The formula a/(n+1) × x^(n+1) is derived from the standard power rule for integration and is valid for all real n except −1. For f(x) = a/x, you must use the natural logarithm: ∫ a/x dx = a·ln|x| + C. This calculator handles polynomial cases only; for n = −1 you would need a different tool or a symbolic integration system.
How do I interpret a negative result from a definite integral?
A negative definite integral means the curve lies predominantly below the x-axis over the integration interval, so the signed area is negative. The integral measures net signed area, not total geometric area. If you need the total (unsigned) area, you must split the interval at every x-intercept and sum the absolute values of each sub-integral. For example, ∫[−1 to 1] x dx = 0, because the negative area from −1 to 0 exactly cancels the positive area from 0 to 1.