Calculus
Double and triple integrals: laboratory
Build the sum one strip or shell at a time. Compare domain, limits, Jacobian, exact value and numerical quadrature in nine guided examples.
Integrand, domain and iterated integral
Build the sum
Each second adds all cells in one strip or shell of normalized outer coordinate u. Step, pause or jump to the end.
Physical domain and 3D contributions
Grid in normalized coordinates (u,v)
One cell of the sum
u,v,w lie in [0,1]. J includes interval rescaling; Δu=Δv=Δw=1/n. The highlighted cell belongs to the current strip (the first before starting).
Outer integrand: exact curve and numerical rectangles
The blue curve is the exact result of inner integrations in normalized coordinates. Rectangles use numerical inner sums, not necessarily the exact curve’s midpoint values.
Compare convergence
Deterministic midpoint quadrature, not a general solver or certified error estimate. Examples are dimensionless. Roundoff may produce tiny nonzero errors. Relative error is undefined for a zero exact integral.