The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Signed integrals and geometric areas
Distinguish a signed integral from nonnegative geometric area.
Find all relevant intersections and split only where the separation changes sign.
Use vertical or horizontal slices and solve area conditions with parameters.
An integral is signed; geometric area is nonnegative. Find roots or intersections before integrating. Split where the sign or upper curve changes. Use upper minus lower for vertical slices, or right minus left for horizontal slices.