∇
VECTOR Calc 1–4
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120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
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Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
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Chapter 5 Theory

Double & Triple Integrals (Rectangular, Cylindrical & Spherical)

3D Riemann prism columns, Fubini's Theorem, and coordinate Jacobians

Adjust the 3D grid partition N×N to see rectangular Riemann columns approximate the volume ∬_R f(x,y) dA under a surface.

1. 3D Riemann Columns & Coordinate Jacobians

Partitioning a 2D region R into Δx × Δy tiles creates vertical 3D columns of volume f(x_i, y_j) Δx Δy. When integrating over circular or spherical regions, switching coordinates stretches area/volume elements by the Jacobian factor r (polar/cylindrical) or ρ² sin(φ) (spherical).

dA=dx dy=r dr dθ,dV=dx dy dz=ρ2sin⁡ϕ dρ dϕ dθdA = dx\,dy = r\,dr\,d\theta, \qquad dV = dx\,dy\,dz = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta

Quick Reference & Cheat Sheet

Cylindrical & Spherical Volume Elements
dVcyl=r dz dr dθ,dVsph=ρ2sin⁡ϕ dρ dϕ dθdV_{\text{cyl}} = r\,dz\,dr\,d\theta, \quad dV_{\text{sph}} = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta
Jacobian determinant accounts for radial/angular wedge expansion.