Definite Integral Formula

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

The Formula

∫abf(x) dx=F(b)βˆ’F(a)\int_a^b f(x) \, dx = F(b) - F(a)

When to use: The signed total area under the curve from aa to bbβ€”positive above the xx-axis, negative below.

Quick Example

∫032x dx=[x2]03=9βˆ’0=9\int_0^3 2x \, dx = [x^2]_0^3 = 9 - 0 = 9 β€” substitute 3 then 0 and subtract.

Notation

∫abf(x) dx\int_a^b f(x)\,dx with lower bound aa and upper bound bb. [F(x)]ab[F(x)]_a^b means F(b)βˆ’F(a)F(b) - F(a).

What This Formula Means

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

The signed total area under the curve from aa to bbβ€”positive above the xx-axis, negative below.

Formal View

∫abf(x) dx=lim⁑βˆ₯Pβˆ₯β†’0βˆ‘i=1nf(xiβˆ—)Ξ”xi\int_a^b f(x)\,dx = \lim_{\|P\| \to 0} \sum_{i=1}^{n} f(x_i^*) \Delta x_i, where PP is a partition of [a,b][a, b] and βˆ₯Pβˆ₯\|P\| is the mesh size. If Fβ€²=fF' = f on [a,b][a,b], then ∫abf(x) dx=F(b)βˆ’F(a)\int_a^b f(x)\,dx = F(b) - F(a).

Worked Examples

Example 1

easy
Evaluate ∫14(2xβˆ’1) dx\int_1^4 (2x - 1)\,dx.

Answer

1212

First step

1
Find the antiderivative: F(x)=x2βˆ’xF(x) = x^2 - x.

Full solution

  1. 2
    Apply the Fundamental Theorem: ∫14(2xβˆ’1) dx=F(4)βˆ’F(1)\int_1^4 (2x-1)\,dx = F(4) - F(1).
  2. 3
    Compute F(4)=16βˆ’4=12F(4) = 16 - 4 = 12 and F(1)=1βˆ’1=0F(1) = 1 - 1 = 0.
  3. 4
    Result: 12βˆ’0=1212 - 0 = 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 βˆ«βˆ’12(x2βˆ’x) dx\int_{-1}^{2} (x^2 - x)\,dx and interpret the sign of the result.

Example 3

medium
Evaluate ∫02(3x2+2x+1) dx\int_0^2 (3x^2 + 2x + 1)\,dx step by step.

Common Mistakes

  • Forgetting that area below the axis is negative β€” the definite integral is signed, so a symmetric curve like sin⁑x\sin x over [0,2Ο€][0,2\pi] integrates to zero.
  • Subtracting in the wrong order β€” it is F(b)βˆ’F(a)F(b)-F(a) (top bound minus bottom bound), not the reverse.
  • Carrying a +C+C into a definite integral β€” the constants cancel in F(b)βˆ’F(a)F(b)-F(a), so drop CC when bounds are present.

Why This Formula Matters

The definite integral is where calculus delivers a usable number: total displacement, total area, accumulated charge. The 'signed' part is the trap and the insight β€” area below the axis subtracts, so net area can be zero even when there's plenty of region, which matters for displacement versus distance. Recognizing it by "Are there bounds aa and bb giving one number that counts area below the axis as negative?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from indefinite integral and total area (unsigned) and riemann sum in a mixed problem set.

Frequently Asked Questions

What is the Definite Integral formula?

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

How do you use the Definite Integral formula?

The signed total area under the curve from aa to bbβ€”positive above the xx-axis, negative below.

What do the symbols mean in the Definite Integral formula?

∫abf(x) dx\int_a^b f(x)\,dx with lower bound aa and upper bound bb. [F(x)]ab[F(x)]_a^b means F(b)βˆ’F(a)F(b) - F(a).

Why is the Definite Integral formula important in Math?

The definite integral is where calculus delivers a usable number: total displacement, total area, accumulated charge. The 'signed' part is the trap and the insight β€” area below the axis subtracts, so net area can be zero even when there's plenty of region, which matters for displacement versus distance. Recognizing it by "Are there bounds aa and bb giving one number that counts area below the axis as negative?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from indefinite integral and total area (unsigned) and riemann sum in a mixed problem set.

What do students get wrong about Definite Integral?

The procedure for definite integral is the easy part; the trap is forgetting that area below the axis is negative. Asking "Are there bounds aa and bb giving one number that counts area below the axis as negative?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 β†’
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