# Mastering Triple Integration: Concepts and Coordinate Systems
> Learn triple integration in rectangular, cylindrical, and spherical coordinates with a step-by-step volume calculation example using Fubini's Theorem.

Tags: calculus, triple-integration, multivariable-calculus, cylindrical-coordinates, spherical-coordinates, fubinis-theorem, math-tutorial
## Triple Integration Overview
* Definition: Integrating a function f(x, y, z) over a 3D bounded region to find hypervolume or total mass.
* Interpretation: Represents mass when density is variable.

## Coordinate Systems
* **Rectangular (x, y, z):** Uses dV = dx dy dz. Ideal for planes and boxes.
* **Cylindrical:** Used for axis symmetry. Transformation: x=r cos(θ), y=r sin(θ), z=z. Jacobian: dV = r dz dr dθ.
* **Spherical:** Ideal for spheres and cones. Volume element: dV = ρ² sin(φ) dρ dφ dθ.

## Fubini's Theorem
* Allows triple integrals to be computed as iterated integrals.
* Order of integration (dx dy dz, etc.) does not change the result if the function is continuous.

## Example Problem & Solution
* **Problem:** Find the volume of a solid bounded by z = 1 - y², z = 0, x = 0, and x = 2.
* **Limits:** x from [0, 2], y from [-1, 1], z from [0, 1-y²].
* **Calculation:** ∫(0 to 2) ∫(-1 to 1) (1 - y²) dy dx.
* **Result:** Total Volume = 8/3.

## Applications
* Calculating mass and center of gravity.
* Computing flux in vector fields.
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