Cylindrical and Spherical

Key idea

Coordinate systems for 3D integration.

In depth

Cylindrical: (r, θ, z) with dV = r dr dθ dz. Spherical: (ρ, φ, θ) with dV = ρ² sin φ dρ dφ dθ. Use for cylinders, cones, spheres where Cartesian limits are messy.

Worked example

Volume of a sphere radius a: ∫₀²π∫₀^π∫₀ᵃ ρ² sin φ dρ dφ dθ = (4/3)πa³.

Formula

Real-world uses

• Solving problems about spheres, cylinders and gravitational fields in physics

Common mistake

Omitting the Jacobian factor ρ² sin φ.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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