# Mastering Double and Triple Integration: Concepts & Examples
> Learn double and triple integration. Covers Fubini's Theorem, coordinate transformations (Polar, Cylindrical, Spherical), and solved volume examples.

Tags: calculus, multivariable-calculus, integration, fubinis-theorem, mathematics, engineering-math, multiple-integrals
## Double and Triple Integration Overview
Developmental guide to multiple integrals for calculating areas, volumes, and physical properties like mass and center of gravity.

## The Double Integral
* Defined over region D in the xy-plane.
* Represents volume under surface z = f(x,y).
* Computed as iterated integrals (dx dy or dy dx).

## Fubini's Theorem
* For continuous functions on rectangular regions, the order of integration does not change the result.

## Solved Example: Double Integration
* Problem: ∫ from 0 to 2 [ ∫ from 0 to 1 (2x + y) dy ] dx
* Step 1: Integrate w.r.t y → 2x + 0.5
* Step 2: Integrate w.r.t x → [x² + 0.5x] from 0 to 2
* Final Result: 5

## Triple Integrals
* Integration over a 3D solid region E: ∭ f(x,y,z) dV.
* Used for volume (where f=1) and variable density/mass calculations.

## Solved Example: Triple Integration
* Function: f(x,y,z) = xyz over a unit cube [0,1].
* Method: Separable Limit Trick.
* Calculation: (1/2) * (1/2) * (1/2) = 1/8.

## Coordinate Systems & Jacobian
* **Polar (2D)**: dA = r dr dθ
* **Cylindrical (3D)**: dV = r dz dr dθ
* **Spherical (3D)**: dV = ρ² sin(φ) dρ dθ dφ
* The **Jacobian** serves as the scale factor to preserve volume during transformation.
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