Last updated: 2026-05-26

Definition

A definite integral measures the signed area between a function and the x-axis over a closed interval $[a, b]$:

$\int_a^b P(x)\,dx$

"Signed" means areas below the x-axis count as negative. When the curve dips below zero in part of the interval, that portion subtracts from the total. The result is a single number, not a function.

The Fundamental Theorem of Calculus

The key insight that makes definite integration practical is the Fundamental Theorem of Calculus (FTC):

$\int_a^b P(x)\,dx = F(b) - F(a)$

where $F(x)$ is any antiderivative of $P(x)$ — a function satisfying $F'(x) = P(x)$.

Instead of summing infinitely many infinitesimal rectangles, the theorem reduces the calculation to three steps:

1. Find the antiderivative $F(x)$.
2. Evaluate $F$ at both limits.
3. Subtract.

The Power Rule for Integration

For polynomials, the antiderivative of each term is found using the power rule:

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \qquad (n \neq -1)$

Apply this term-by-term to integrate any polynomial exactly.

Example — ∫₀² (3x² − 1) dx:

Term	Power rule	Antiderivative term
$3x^2$	$3 \cdot \dfrac{x^3}{3}$	$x^3$
$-1$	$-1 \cdot \dfrac{x^1}{1}$	$-x$

So $F(x) = x^3 - x$.

$\int_0^2 (3x^2 - 1)\,dx = F(2) - F(0) = (8 - 2) - (0 - 0) = 6$

How to Enter Coefficients

Enter polynomial coefficients from the highest degree to the lowest, separated by commas.

Polynomial	Coefficients input
$x^2$	1, 0, 0
$3x^2 - 1$	3, 0, -1
$2x^3 + x - 5$	2, 0, 1, -5
$7$ (constant)	7

The number of values determines the degree: four values → degree 3 polynomial.

Signed Area and Reversing Limits

The definite integral gives signed area, which has two important consequences:

When the curve is below the x-axis: contributions are negative. For example, $\int_0^1 (-x)\,dx = -\frac{1}{2}$, even though the geometric region has area $\frac{1}{2}$.

When $a > b$: the integral is the negative of integrating from $b$ to $a$:

$\int_a^b P(x)\,dx = -\int_b^a P(x)\,dx$

This is not an error — it is a fundamental property of the integral. The calculator handles this correctly; entering $a = 2, b = 0$ gives the negation of $a = 0, b = 2$.

Constant of Integration

An indefinite integral is written $F(x) + C$, where $C$ is an arbitrary constant. For definite integrals, $C$ cancels:

$[F(x) + C]_a^b = (F(b) + C) - (F(a) + C) = F(b) - F(a)$

So the constant of integration is irrelevant for definite integrals and is omitted from the antiderivative shown here.

When to Use Numerical Integration

The power rule works only for polynomials ($x^n$ with integer $n \geq 0$). Other functions require numerical methods:

Function type	Method
$\sin x$, $\cos x$, $e^x$	Known antiderivatives (exact)
Rational, algebraic	Partial fractions, substitution
No closed-form antiderivative	Simpson's rule, Gaussian quadrature

When the integrand cannot be written as a polynomial, this calculator does not apply — a numerical integration method or a computer algebra system handles those cases.

Practical Applications

Definite integrals appear throughout science and engineering:

• Physics: displacement from velocity ($\int v\,dt$), work done by a force ($\int F\,dx$).
• Economics: consumer surplus (area between demand curve and price level).
• Probability: the probability that a continuous random variable falls in $[a, b]$ is $\int_a^b f(x)\,dx$ where $f$ is the density function.
• Geometry: area between two curves, volume of solids of revolution.

Frequently Asked Questions (FAQ)

Can I integrate any function with this calculator?

This calculator handles polynomials only — functions of the form P(x) = aₙxⁿ + … + a₁x + a₀. For these, the antiderivative is found term-by-term using the power rule, giving an exact result with no approximation error. Non-polynomial integrands (sin, eˣ, ln x, etc.) require numerical integration methods such as Simpson’s rule or Gaussian quadrature.

What happens when the lower limit is greater than the upper limit (a > b)?

The calculator still gives the correct signed area. By the property of definite integrals, ∫_a^b P(x) dx = −∫_b^a P(x) dx. So if a > b the result will simply be the negative of integrating from b to a. This is mathematically valid and useful when setting up double integrals or working with flux integrals.

What is the antiderivative and why does it help?

The antiderivative F(x) of P(x) is a function whose derivative is P(x). For a monomial aₙxⁿ, the power rule gives aₙxⁿ⁺¹/(n+1). To integrate a polynomial, apply the rule to each term and sum the results. The Fundamental Theorem of Calculus then says ∫_a^b P(x) dx = F(b) − F(a), turning the integration problem into simple substitution.

Why is the constant of integration not shown?

When evaluating a definite integral, the constant of integration C cancels out: [F(x) + C] evaluated from a to b is (F(b) + C) − (F(a) + C) = F(b) − F(a). The C doesn't affect the answer, so it's conventionally omitted. The constant only matters for indefinite integrals, where the family of antiderivatives is written F(x) + C.