_abf(x),dx is signed; total geometric area adds the positive sizes of all pieces. A vertical strip has positive height above the axis and negative height below it in the signed integral. Geometric area uses a non-negative height. Solve f(x)=0 in the interval; check the sign between successive zeros; integrate f above the axis and -f below, then add. A touching zero does not reverse the sign. _02(x-1),dx=0 but A=- _01(x-1),dx+ _12(x-1),dx=1. Exam check: taking the absolute value of a final signed integral fails when positive and negative contributions cancel.

\(\int_a^bf(x)\,\mathrm{d}x\) is signed; total geometric area adds the positive sizes of all pieces.

A vertical strip has positive height above the axis and negative height below it in the signed integral. Geometric area uses a non-negative height.

Solve \(f(x)=0\) in the interval; check the sign between successive zeros; integrate \(f\) above the axis and \(-f\) below, then add. A touching zero does not reverse the sign.

\(\int_0^2(x-1)\,\mathrm{d}x=0\) but \(A=-\int_0^1(x-1)\,\mathrm{d}x+\int_1^2(x-1)\,\mathrm{d}x=1\).

Exam check: taking the absolute value of a final signed integral fails when positive and negative contributions cancel.

Signed integrals and geometric areas

Signed integrals and geometric areas