Calculus

Integral

An integral is a calculus operation used to accumulate quantities. A definite integral can represent the signed area between a graph and the x-axis over an interval.

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Meaning

What Is Integral?

An integral is a calculus operation used to accumulate quantities. A definite integral can represent the signed area between a graph and the x-axis over an interval.

Examples

Examples of Integral

1∫x dx = x²/2 + C

2∫2x dx = x² + C

3∫₀² x dx = 2

Understand

Formula and Key Points

Formula / rule

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1

Indefinite integrals include a constant of integration C.
Definite integrals have lower and upper limits.
Integration reverses differentiation in many situations.
Integrals are used for area, volume and accumulated change.

CSE Connection

Why This Matters in Computer Science

Used in optimization, simulation, graphics, machine learning, control systems and numerical computing.

Step by Step

Worked Example

Problem

Find \int 2x dx.

Increase the power of x from 1 to 2.
Divide by the new power: 2 × x²/2.
Simplify to x².
Add the constant of integration C.

✓

Final Answer

x² + C

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Compare

Derivative vs Integral

Feature	Derivative	Integral
Main idea	Rate of change	Accumulation
Basic action	Differentiation	Integration
Typical geometry	Slope of tangent	Area / accumulated quantity

FAQ

Integral: Frequently Asked Questions

What is an integral used for?

Integrals are used to calculate accumulated quantities such as area, displacement and volume.

Why do indefinite integrals use + C?

Different functions can have the same derivative, so the constant represents that family of antiderivatives.