Horizontal Integration

Calculus 2 · Axiom Academy

When integrating with respect to y turns a two-piece area problem into a single clean integral. 1. The Standard Picture: Vertical Strips Integrating with respect to x slices the region into thin vertical strips . Each strip is anchored on the x -axis and reaches up from the bottom boundary to the top boundary. Width = dx , an infinitesimal step in x You sum strips left to right across the x -axis Vertical strips: integrate over x Now integrate with respect to y . The same region is sliced into horizontal strips , each anchored on the y -axis and reaching from the left boundary across to the right boundary. Width = dy , an infinitesimal step in y You sum strips bottom to top up the y -axis Horizontal strips: integrate over y Watch a single region filled both ways at once. On the left, vertical strips march across in x ; on the right, horizontal strips climb in y . Both tile the same area — the only question is which sweep gives a simpler integral. Simple when the region has one top curve and one bottom curve the whole way across. Simple when the region has one right curve and one left curve from bottom to top. 4. The Test: Does a Strip Stay Simple?

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