∫∫
Double Integrals Volume
Calculus III · Multivariable Integration
Learning Objectives
Volume = ∫∫ f(x,y) dA as sum of prisms
Riemann sum approximation with n² boxes
Iterated integrals over rectangular regions
Convergence as n→∞
Formula
Volume:
V = ∫∫_R f(x,y) dA
Riemann Sum:
V ≈ Σᵢ Σⱼ f(xᵢ, yⱼ) ΔA
where ΔA = Δx · Δy
Tags
double integral
volume
Riemann sum
3D
SURFACE f(x,y)
Dome: 4 − x² − y²
Plane: 3 − x − y
Wavy: 1 + sin(πx)·sin(πy)
Cone: 2 − √(x²+y²)
Subdivisions n
6
Riemann Sum ≈ Volume
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Exact Value
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Error
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Each translucent column is f(xᵢ,yⱼ)·ΔA. More subdivisions → more columns → closer to exact volume.
Drag the surface to rotate. Increase n to see convergence.