Definite Integral Formula

The Formula

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

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

Quick Example

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

Notation

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

What This Formula Means

An integral evaluated between specific lower and upper bounds, yielding a single numerical value rather than a function.

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

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).

Worked Examples

Example 1

easy
Evaluate \int_1^4 (2x - 1)\,dx.

Solution

  1. 1
    Find the antiderivative: F(x) = x^2 - x.
  2. 2
    Apply the Fundamental Theorem: \int_1^4 (2x-1)\,dx = F(4) - F(1).
  3. 3
    Compute F(4) = 16 - 4 = 12 and F(1) = 1 - 1 = 0.
  4. 4
    Result: 12 - 0 = 12.

Answer

12
For a definite integral, find the antiderivative, evaluate it at the upper bound, then subtract its value at the lower bound. The constant of integration cancels out when you subtract, so it can be omitted.

Example 2

medium
Evaluate \int_{-1}^{2} (x^2 - x)\,dx and interpret the sign of the result.

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.

Why This Formula Matters

Computes exact areas, distances, and accumulated quantities.

Frequently Asked Questions

What is the Definite Integral formula?

An integral evaluated between specific lower and upper bounds, yielding a single numerical value rather than a function.

How do you use the Definite Integral formula?

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

What do the symbols mean in the Definite Integral formula?

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

Why is the Definite Integral formula important in Math?

Computes exact areas, distances, and accumulated quantities.

What do students get wrong about Definite Integral?

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

What should I learn before the Definite Integral formula?

Before studying the Definite Integral formula, you should understand: integral.

Want the Full Guide?

This formula is covered in depth in our complete guide:

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