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Definite Integral Calculator

Calculate the exact value of any definite integral with step-by-step solutions and shaded area visualization.

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Basic Polynomial

Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.

Key values: x^2 · [0, 1] · Simpson's rule

Trigonometric Integral

Integrate sin(x) from 0 to pi. Exact answer: 2.

Key values: sin(x) · [0, pi] · Exact: 2

Area Under a Bell Curve

Approximate the Gaussian integral e^(-x^2) from -3 to 3.

Key values: e^(-x^2) · [-3, 3] · Gaussian

Documentation

The Definite Integral

The definite integral of $f(x)$ from $a$ to $b$ accumulates the signed area between the curve and the x-axis:

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

where $F$ is any antiderivative of $f$ . This is the Fundamental Theorem of Calculus — it connects differentiation and integration.

Key Properties

Property	Formula
Linearity	$\int_a^b [cf(x) + g(x)]\,dx = c\!\int_a^b f\,dx + \int_a^b g\,dx$
Reversing limits	$\int_a^b f\,dx = -\int_b^a f\,dx$
Splitting interval	$\int_a^b f\,dx = \int_a^c f\,dx + \int_c^b f\,dx$
Zero-width	$\int_a^a f\,dx = 0$
Comparison	If $f(x) \geq 0$ on $[a,b]$ , then $\int_a^b f\,dx \geq 0$

Evaluation Example

Compute: $\int_1^3 (2x^2 + 1) \, dx$

= \left[\frac{2x^3}{3} + x\right]_1^3 = \left(\frac{54}{3} + 3\right) - \left(\frac{2}{3} + 1\right) = 21 - \frac{5}{3} = \frac{58}{3}

Signed vs. Total Area

The definite integral gives signed area — regions below the x-axis contribute negatively. For the total area(always positive), integrate the absolute value:

\text{Total area} = \int_a^b |f(x)| \, dx

In practice, find where $f(x) = 0$ , split at those points, and negate the integral on intervals where $f$ is negative.

Frequently Asked Questions

What is a definite integral?

A definite integral $\int_a^b f(x)\\,dx$ computes the signed area between the curve $f(x)$ and the x-axis over the interval $[a, b]$ . By the Fundamental Theorem of Calculus, it equals $F(b) - F(a)$ , where $F$ is any antiderivative of $f$ .

What is the difference between signed area and total area?

Signed area counts regions below the x-axis as negative, so they subtract from the total. Total area uses $|f(x)|$ so all regions contribute positively. For example, for $\sin(x)$ from $0$ to $2\pi$ , the signed area is 0 but the total area is 4.

What does the Fundamental Theorem of Calculus say?

The Fundamental Theorem connects differentiation and integration. It states that if $F$ is an antiderivative of $f$ , then the definite integral from $a$ to $b$ of $f(x)\\,dx$ equals $F(b) - F(a)$ . This transforms the problem of computing area into evaluating a function at two points.

Can a definite integral be negative?

Yes. When $f(x)$ is below the x-axis over most of the interval, the integral is negative because the signed area below the axis outweighs the area above it.

What happens when the upper and lower bounds are reversed?

Reversing the bounds negates the integral: $\int_b^a f(x)\\,dx = -\int_a^b f(x)\\,dx$ . This is one of the fundamental properties of definite integrals.

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