Definite integrals, areas and volumes
Definite integrals, areas and volumes · Diagrams
Signed integrals and areas
Equal areas, opposite signed integrals
Use the source sine intervals: [−π, 0], [0, π] and [−π, π]. Compare the signed integral with the total geometric area. Decimal axis labels approximate the exact preset bounds.
Teaching and source notes
The source sine regions each have geometric area 2. Their signed integrals are −2 and +2, so the combined signed integral is 0 while the geometric area is 4. The explorer retains those exact preset intervals in radians.
For a region above the axis, area is the integral. Below the axis, area is the negative of the integral. Across a sign change, split the interval and add the nonnegative areas; taking the absolute value of the combined integral can lose area.
Shaded running areas are numerical estimates, compared with the antiderivative −cos x. Rounded numerical agreement does not replace the analytic evaluation or the sign split.