Volume Spinner

Approximate integrals, rotate regions into solids, and master integration techniques.

Numerical laboratory

Estimate—

Error—

Left Midpoint Trapezium

The maths

Numerical rules: left Lₙ = hΣf(a + ih); midpoint Mₙ = hΣf(a + (i + ½)h); trapezium Tₙ = h[f(a)/2 + Σf(a + ih) + f(b)/2], where h = (b − a)/n.

For a concave-up graph, the trapezium rule overestimates the signed integral; for concave-down it underestimates.

Rotation about x: V = π∫[f(x)]² dx, using discs.

Rotation about y: V = 2π∫x f(x) dx for non-negative f(x), using cylindrical shells.

Substitution: replace a composite expression with u and transform du and the limits.

By parts: ∫u dv = uv − ∫v du. Volumes use units³; areas use units².

A solid is built from thin cross-sections.