Lab · Definite integrals
Definite integrals and rotations: lab
Choose what to measure: signed integral, plane area, volume or generated surface area. The formula depends on the geometry.
Choose a case and quantity, or enter your formulas.
Numerical result with n subdivisions
With 2n subdivisions
Difference |Q₂ₙ−Qₙ|, not a guaranteed error
Exact reference for the guided case
Added base area:
Signed integration respects a→b. Geometric measurements always use [min(a,b),max(a,b)].
This is a numerical check, not a general proof of integrability or domain validity. Sampling and refinement may miss peaks, oscillations or very close intersections. Free geometric checks are precautionary, not symbolic.
Guided case: f√(1+f′²)=1 in the interior; endpoints use its continuous limit, avoiding 0·∞.
The local contribution uses the midpoint: ΔQ≈h(x*)Δx. It is not a Simpson or trapezoidal weight.
Plane region and selected layer
Building the rotation
The preview may show a partial turn; numerical results always refer to 360°. This is an illustration, not a second measurement.
Show explanation and steps
How the integrand is built
Comparison as the partition is refined
Results follow the selected quadrature. Observed convergence does not certify the error.
Read the definite integrals and rotations article · Review antiderivatives