Definite Integral

Calculus
definition

Also known as: area

Grade 9-12

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An integral evaluated between specific bounds a and b, yielding a single number: the signed area under the curve. Computes exact areas, total distances traveled, accumulated quantities, and work done by a force.

This concept is covered in depth in our Integration of Rational Functions Guide, with worked examples, practice problems, and common mistakes.

Definition

An integral evaluated between specific bounds a and b, yielding a single number: the signed area under the curve.

💡 Intuition

The signed total area under the curve from a to b—positive above the x-axis, negative below.

🎯 Core Idea

Definite integral gives a number; indefinite integral gives a function.

Example

\int_0^3 2x \, dx = [x^2]_0^3 = 9 - 0 = 9 — substitute 3 then 0 and subtract.

Formula

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

Notation

\int_a^b f(x)\,dx with lower bound a and upper bound b. [F(x)]_a^b means F(b) - F(a).

🌟 Why It Matters

Computes exact areas, total distances traveled, accumulated quantities, and work done by a force. The definite integral is what makes calculus useful in physics, engineering, and economics — it turns rate-of-change information back into totals.

💭 Hint When Stuck

Write out F(b) - F(a) step by step, substituting each bound separately before subtracting.

Formal View

\int_a^b f(x)\,dx = \lim_{\|P\| \to 0} \sum_{i=1}^{n} f(x_i^*) \Delta x_i, where P is a partition of [a, b] and \|P\| is the mesh size. If F' = f on [a,b], then \int_a^b f(x)\,dx = F(b) - F(a).

🚧 Common Stuck Point

Area below the x-axis counts as negative — if you need total geometric area, integrate the absolute value.

⚠️ Common Mistakes

  • Evaluating F(a) - F(b) instead of F(b) - F(a) — the upper bound goes first: \int_a^b f(x)\,dx = F(b) - F(a).
  • Treating the definite integral as always giving positive area: \int_0^{\pi} \sin x \, dx = 2, but \int_{\pi}^{2\pi} \sin x \, dx = -2 because the curve is below the x-axis.
  • Forgetting that swapping the limits of integration changes the sign: \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx.

Frequently Asked Questions

What is Definite Integral in Math?

An integral evaluated between specific bounds a and b, yielding a single number: the signed area under the curve.

What is the Definite Integral formula?

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

When do you use Definite Integral?

Write out F(b) - F(a) step by step, substituting each bound separately before subtracting.

Prerequisites

integral

How Definite Integral Connects to Other Ideas

To understand definite integral, you should first be comfortable with integral. Once you have a solid grasp of definite integral, you can move on to fundamental theorem.

Want the Full Guide?

This concept is explained step by step in our complete guide:

How to Integrate Rational Functions: Long Division and Partial Fractions →
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Visualization

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Visual representation of Definite Integral